Introduction to Video: Normal Approximation of the Binomial and Poisson Distributions; 00:00:34 â How to use the normal distribution as an approximation for the binomial or poisson with â¦ For large value of the λ (mean of Poisson variate), the Poisson distribution can be well approximated by a normal distribution with the same mean and variance. There are some properties of the Poisson distribution: To calculate the Poisson distribution, we need to know the average number of events. The Poisson distribution can also be used for the number of events in other intervals such as distance, area or volume. Poisson Approximation of Binomial Probabilities. Verify whether n is large enough to use the normal approximation by checking the two appropriate conditions.. For the above coin-flipping question, the conditions are met because n â p = 100 â 0.50 = 50, and n â (1 â p) = 100 â (1 â 0.50) = 50, both of which are at least 10.So go ahead with the normal approximation. The mean of Poisson random variable X is Î¼ = E (X) = Î» and variance of X is Ï 2 = V (X) = Î». This approximates the binomial probability (with continuity correction) and graphs the normal pdf over the binomial pmf. That is $Z=\frac{X-\mu}{\sigma}=\frac{X-\lambda}{\sqrt{\lambda}} \sim N(0,1)$. For sufficiently large Î», X â¼ N (Î¼, Ï 2). Understand Poisson parameter roughly. For instance, the Poisson distribution calculator can be applied in the following situations: The probability of a certain number of occurrences is derived by the following formula: $$P(X=x)=\frac{e^{-\lambda}\lambda^x}{x! To analyze our traffic, we use basic Google Analytics implementation with anonymized data. It represents the probability of some number of events occurring during some time period. Comment/Request I was expecting not only chart visualization but a numeric table. Clearly, Poisson approximation is very close to the exact probability. P ... where n is closer to 300, the normal approximation is as good as the Poisson approximation. a. Step by Step procedure on how to use normal approximation to poission distribution calculator with the help of examples guide you to understand it. The general rule of thumb to use normal approximation to Poisson distribution is that Î» is sufficiently large (i.e., Î» â¥ 5). 28.2 - Normal Approximation to Poisson Just as the Central Limit Theorem can be applied to the sum of independent Bernoulli random variables, it can be applied to â¦ Therefore, we plug those numbers into the Poisson Calculator and hit the Calculate button. The value of average rate must be positive real number while the value of Poisson random variable must positive integers. Poisson Probability Calculator. Doing so, we get: The Poisson Probability Calculator can calculate the probability of an event occurring in a given time interval. Find what is poisson distribution for given input data? In case you have any suggestion, or if you would like to report a broken solver/calculator, please do not hesitate to contact us. Solution : f(x, λ) = 2.58 x e-2.58! Examples. q = 1 - p M = N x p SD = â (M x q) Z Score = (x - M) / SD Z Value = (x - M - 0.5)/ SD Where, N = Number of Occurrences p = Probability of Success x = Number of Success q = Probability of Failure M = Mean SD = Standard Deviation Approximate the probability that. Normal approximation to the binomial distribution. Normal Approximation for the Poisson Distribution Calculator More about the Poisson distribution probability so you can better use the Poisson calculator above: The Poisson probability is a type of discrete probability distribution that can take random values on the range [0, +\infty) [0,+â). When we are using the normal approximation to Binomial distribution we need to make correction while calculating various probabilities. That is Z=\dfrac{X-\lambda}{\sqrt{\lambda}}\to N(0,1) for large \lambda. The sum of two Poisson random variables with parameters Î»1 and Î»2 is a Poisson random variable with parameter Î» = Î»1 + Î»2. 2.1.6 More on the Gaussian The Gaussian distribution is so important that we collect some properties here. Press the " GENERATE WORK " button to make the computation. Let X be a Poisson distributed random variable with mean \lambda. Calculate nq to see if we can use the Normal Approximation: Since q = 1 - p, we have n(1 - p) = 10(1 - 0.4) nq = 10(0.6) nq = 6 Since np and nq are both not greater than 5, we cannot use the Normal Approximation to the Binomial Distribution.cannot use the Normal Approximation to the Binomial Distribution. P (Y â¥ 9) = 1 â P (Y â¤ 8) = 1 â 0.792 = 0.208 Now, let's use the normal approximation to the Poisson to calculate an approximate probability. As Î» increases the distribution begins to look more like a normal probability distribution. The probability that on a given day, at least 65 kidney transplants will be performed is,$$ \begin{aligned} P(X\geq 65) &= 1-P(X\leq 64)\\ &= 1-P(X\leq 64.5)\\ & \quad\quad (\text{Using continuity correction})\\ &= 1-P\bigg(\frac{X-\lambda}{\sqrt{\lambda}} < \frac{64.5-45}{\sqrt{45}}\bigg)\\ &= 1-P(Z\leq 3.06)\\ &= 1-0.9989\\ & \quad\quad (\text{Using normal table})\\ &= 0.0011 \end{aligned} $$, c. The probability that on a given day, no more than 40 kidney transplants will be performed is,$$ \begin{aligned} P(X < 40) &= P(X < 39.5)\\ & \quad\quad (\text{Using continuity correction})\\ &= P\bigg(\frac{X-\lambda}{\sqrt{\lambda}} < \frac{39.5-45}{\sqrt{45}}\bigg)\\ &= P(Z < -0.82)\\ & = P(Z < -0.82) \\ &= 0.2061\\ & \quad\quad (\text{Using normal table}) \end{aligned} $$. The value of average rate must be positive real number while the value of Poisson random variable must positive integers. Thus \lambda = 69 and given that the random variable X follows Poisson distribution, i.e., X\sim P(69). Note that the conditions of Poisson approximation to Binomial are complementary to the conditions for Normal Approximation of Binomial Distribution. \lambda = 45. Normal Approximation to Poisson Distribution Calculator Normal distribution can be used to approximate the Poisson distribution when the mean of Poisson random variable is sufficiently large.When we are using the normal approximation to Poisson distribution we need to make correction while calculating various probabilities. The Poisson distribution tables usually given with examinations only go up to Î» = 6. Poisson Approximation for the Binomial Distribution â¢ For Binomial Distribution with large n, calculating the mass function is pretty nasty â¢ So for those nasty âlargeâ Binomials (n â¥100) and for small Ï (usually â¤0.01), we can use a Poisson with Î» = nÏ (â¤20) to approximate it! If you continue without changing your settings, we'll assume that you are happy to receive all cookies on the vrcacademy.com website. The probability that on a given day, exactly 50 kidney transplants will be performed is,$$ \begin{aligned} P(X=50) &= P(49.5< X < 50.5)\\ & \quad\quad (\text{Using continuity correction})\\ &= P\bigg(\frac{49.5-45}{\sqrt{45}} < \frac{X-\lambda}{\sqrt{\lambda}} < \frac{50.5-45}{\sqrt{45}}\bigg)\\ &= P(0.67 < Z < 0.82)\\ & = P(Z < 0.82) - P(Z < 0.67)\\ &= 0.7939-0.7486\\ & \quad\quad (\text{Using normal table})\\ &= 0.0453 \end{aligned} $$, b. If the number of trials becomes larger and larger as the probability of successes becomes smaller and smaller, then the binomial distribution becomes the Poisson distribution. a specific time interval, length, volume, area or number of similar items). Normal Approximation â Lesson & Examples (Video) 47 min. Step 2:X is the number of actual events occurred. For sufficiently large values of Î», (say Î»>1,000), the Normal(Î¼ = Î»,Ï2= Î»)Distribution is an excellent approximation to the Poisson(Î»)Distribution. Thus, withoutactually drawing the probability histogram of the Poisson(1) we know that it is strongly skewed to the right; indeed, it has no left tail! There is a less commonly used approximation which is the normal approximation to the Poisson distribution, which uses a similar rationale than that for the Poisson distribution. Question is as follows: In a shipment of 20 engines, history shows that the probability of any one engine proving unsatisfactory is 0.1. The general rule of thumb to use normal approximation to Poisson distribution is that \lambda is sufficiently large (i.e., \lambda \geq 5).eval(ez_write_tag([[468,60],'vrcacademy_com-medrectangle-3','ezslot_1',126,'0','0'])); For sufficiently large \lambda, X\sim N(\mu, \sigma^2). To understand more about how we use cookies, or for information on how to change your cookie settings, please see our Privacy Policy. A random sample of 500 drivers is selected. If Î» is greater than about 10, then the Normal Distribution is a good approximation if an appropriate continuity correctionis performed. The FAQ may solve this. A radioactive element disintegrates such that it follows a Poisson distribution. Below is the step by step approach to calculating the Poisson distribution formula. ... Then click the 'Calculate' button. Generally, the value of e is 2.718. Poisson approximations 9.1Overview The Bin(n;p) can be thought of as the distribution of a sum of independent indicator random variables X 1 + + X n, with fX i= 1gdenoting a head on the ith toss of a coin that lands heads with probability p. Each X i has a Ber(p) â¦ 13.1.1 The Normal Approximation to the Poisson Please look at the Poisson(1) probabilities in Table 13.1. Enter an average rate of success and Poisson random variable in the box. ... (Exact Binomial Probability Calculator), and np<5 would preclude use the normal approximation (Binomial z-Ratio Calculator). Step 1: e is the Eulerâs constant which is a mathematical constant. Less than 60 particles are emitted in 1 second. You want to calculate the probability (Poisson Probability) of a given number of occurrences of an event (e.g. If the mean number of particles (\alpha) emitted is recorded in a 1 second interval as 69, evaluate the probability of: a. The Poisson distribution uses the following parameter. First, we have to make a continuity correction. Continuity Correction for normal approximation Binomial distribution is a discrete distribution, whereas normal distribution is a continuous distribution. },\quad x=1,2,3,\ldots$$, $$P(k\;\mbox{events in}\; t\; \mbox {interval}\;X=x)=\frac{e^{-rt}(rt)^k}{k! The mean number of \alpha-particles emitted per second 69. The parameter Î» is also equal to the variance of the Poisson distribution. When we are using the normal approximation to Poisson distribution we need to make correction while calculating various probabilities. Below we will discuss some numerical examples on Poisson distribution where normal approximation is applicable. Let X denote the number of kidney transplants per day. customers entering the shop, defectives in a box of parts or in a fabric roll, cars arriving at a tollgate, calls arriving at the switchboard) over a continuum (e.g. Before using the calculator, you must know the average number of times the event occurs in â¦ Estimate if given problem is indeed approximately Poisson-distributed. a. exactly 50 kidney transplants will be performed. Poisson distribution calculator will estimate the probability of a certain number of events happening in a given time. Since \lambda= 69 is large enough, we use normal approximation to Poisson distribution. Formula : This website uses cookies to ensure you get the best experience on our site and to provide a comment feature. The Binomial distribution can be approximated well by Poisson when n is large and p is small with np < 10, as stated x = 0,1,2,3â¦ Step 3:Î» is the mean (average) number of events (also known as âParameter of Poisson Distribution). Poisson (100) distribution can be thought of as the sum of 100 independent Poisson (1) variables and hence may be considered approximately Normal, by the central limit theorem, so Normal (Î¼ = rate*Size = Î» * N, Ï =â (Î»*N)) approximates Poisson (Î» * N = 1*100 = 100). Enter an average rate of success and Poisson random variable in the box. It is necessary to follow the next steps: The Poisson distribution is a probability distribution. Since \lambda= 45 is large enough, we use normal approximation to Poisson distribution. The probability that between 65 and 75 particles (inclusive) are emitted in 1 second is,$$ \begin{aligned} P(65\leq X\leq 75) &= P(64.5 < X < 75.5)\\ & \quad\quad (\text{Using continuity correction})\\ &= P\bigg(\frac{64.5-69}{\sqrt{69}} < \frac{X-\lambda}{\sqrt{\lambda}} < \frac{75.5-69}{\sqrt{69}}\bigg)\\ &= P(-0.54 < Z < 0.78)\\ &= P(Z < 0.78)- P(Z < -0.54) \\ &= 0.7823-0.2946\\ & \quad\quad (\text{Using normal table})\\ &= 0.4877 \end{aligned} $$, © VrcAcademy - 2020About Us | Our Team | Privacy Policy | Terms of Use. It's an online statistics and probability tool requires an average rate of success and Poisson random variable to find values of Poisson and cumulative Poisson distribution. Poisson distribution is a discrete distribution, whereas normal distribution is a continuous distribution. Poisson Distribution = 0.0031. b. Poisson Approximation to Binomial is appropriate when: np < 10 and . c. no more than 40 kidney transplants will be performed. To enter a new set of values for n, k, and p, click the 'Reset' button. The mean of X is \mu=E(X) = \lambda and variance of X is \sigma^2=V(X)=\lambda. We can also calculate the probability using normal approximation to the binomial probabilities. a) Use the Binomial approximation to calculate the Between 65 and 75 particles inclusive are emitted in 1 second. Gaussian approximation to the Poisson distribution. The probability of a certain number of occurrences is derived by the following formula: Poisson distribution is important in many fields, for example in biology, telecommunication, astronomy, engineering, financial sectors, radioactivity, sports, surveys, IT sectors, etc to find the number of events occurred in fixed time intervals. If $$Y$$ denotes the number of events occurring in an interval with mean $$\lambda$$ and variance $$\lambda$$, and $$X_1, X_2,\ldots, X_\ldots$$ are independent Poisson random variables with mean 1, then the sum of $$X$$'s is a Poisson random variable with mean $$\lambda$$. Normal Approximation to Poisson is justified by the Central Limit Theorem. X follows Poisson distribution, i.e., X\sim P(45). For sufficiently large values of Î», (say Î»>1000), the normal distribution with mean Î» and variance Î» (standard deviation ) is an excellent approximation to the Poisson distribution. That is Z = X â Î¼ Ï = X â Î» Î» â¼ N (0, 1). If you take the simple example for calculating Î» => â¦ Input Data : Suppose that only 40% of drivers in a certain state wear a seat belt. It is normally written as p(x)= 1 (2Ï)1/2Ï e â(x µ)2/2Ï2, (50) 7Maths Notes: The limit of a function like (1 + Î´)Î»(1+Î´)+1/2 with Î» # 1 and Î´  1 can be found by taking the It can have values like the following. The calculator reports that the Poisson probability is 0.168. b. at least 65 kidney transplants will be performed, and The normal approximation to the Poisson distribution. The probability that less than 60 particles are emitted in 1 second is,$$ \begin{aligned} P(X < 60) &= P(X < 59.5)\\ & \quad\quad (\text{Using continuity correction})\\ &= P\bigg(\frac{X-\lambda}{\sqrt{\lambda}} < \frac{59.5-69}{\sqrt{69}}\bigg)\\ &= P(Z < -1.14)\\ & = P(Z < -1.14) \\ &= 0.1271\\ & \quad\quad (\text{Using normal table}) \end{aligned} $$, b. Normal Approximation to Poisson The normal distribution can be approximated to the Poisson distribution when Î» is large, best when Î» > 20. That is the probability of getting EXACTLY 4 school closings due to snow, next winter. Approximating a Poisson distribution to a normal distribution. Find the probability that on a given day. λ (Average Rate of Success) = 2.5 However my problem appears to be not Poisson but some relative of it, with a random parameterization. The mean number of kidney transplants performed per day in the United States in a recent year was about 45. The mean number of kidney transplants performed per day in the United States in a recent year was about 45. Objective : (We use continuity correction), a. Binomial probabilities can be a little messy to compute on a calculator because the factorials in the binomial coefficient are so large. Normal Approximation Calculator Example 3. Since the schools have closed historically 3 days each year due to snow, the average rate of success is 3. Translate the problem into a probability statement about X. Poisson distribution calculator calculates the probability of given number of events that occurred in a fixed interval of time with respect to the known average rate of events occurred. Let X denote the number of particles emitted in a 1 second interval. This value is called the rate of success, and it is usually denoted by \lambda. = 125.251840320 = 1525.8789 x 0.08218 x 7 x 6 x 5 x 4 x 3 x 2 x 1 Use Normal Approximation to Poisson Calculator to compute mean,standard deviation and required probability based on parameter value,option and values. When the value of the mean X (Poisson Random Variable) = 8 Poisson Approximation to Binomial Distribution Calculator, Karl Pearson coefficient of skewness for grouped data, Normal Approximation to Poisson Distribution, Normal Approximation to Poisson Distribution Calculator. }$$, By continuing with ncalculators.com, you acknowledge & agree to our, Negative Binomial Distribution Calculator, Cumulative Poisson Distribution Calculator. Normal distribution can be used to approximate the Poisson distribution when the mean of Poisson random variable is sufficiently large.When we are using the normal approximation to Poisson distribution we need to make correction while calculating various probabilities. The plot below shows the Poisson distribution (black bars, values between 230 and 260), the approximating normal density curve (blue), and the second binomial approximation (purple circles). Step 4 - Click on “Calculate” button to calculate normal approximation to poisson. Now, we can calculate the probability of having six or fewer infections as. a. exactly 215 drivers wear a seat belt, b. at least 220 drivers wear a seat belt, The experiment consists of events that will occur during the same time or in a specific distance, area, or volume; The probability that an event occurs in a given time, distance, area, or volume is the same; to find the probability distribution the number of trains arriving at a station per hour; to find the probability distribution the number absent student during the school year; to find the probability distribution the number of visitors at football game per month. We see that P(X = 0) = P(X = 1) and as x increases beyond 1, P(X =x)decreases. Analytics implementation with anonymized data, area or volume a specific time interval of values for,! Examples guide you to understand it Calculator ), and c. no than! During some time period a mathematical constant sufficiently large Î » is also equal to the Binomial approximation to are! Of similar items normal approximation to poisson calculator how to use normal approximation to calculate normal approximation to Poisson is justified by the Limit! Steps: the Poisson distribution where normal approximation to calculate normal approximation ( Binomial z-Ratio )... Random variable in the United States in a recent year was about 45 happy to receive all on. We use basic Google Analytics implementation with anonymized data required probability based on parameter value, option values! $\lambda$ 'll assume that you are happy to receive all cookies on the the! Due to snow, next winter on “ calculate ” button to make correction while calculating various probabilities Binomial! Happening in a 1 second denoted by $\lambda$... where N closer. Â Î » is also equal to the Binomial approximation to Binomial are complementary to Exact. Values for N, k, and it is usually denoted by ! A Poisson distributed random variable must positive integers p... where N is closer to 300, the approximation. X â¼ N ( Î¼, Ï 2 ) the 'Reset ' button volume, area volume. Good as the Poisson distribution when we are using the normal normal approximation to poisson calculator to Poisson distribution a! Probability ( Poisson probability is 0.168 happy to receive all cookies on the Gaussian distribution so... A continuity correction for normal approximation to Binomial distribution x 6 x 5 x x! K, and p, click the 'Reset ' button the Calculator reports that the for. To know the average number of events in other intervals such as distance, area or.! Average number of occurrences of an event occurring in a certain state wear a seat belt will estimate the (. < 10 and no more than 40 kidney transplants per day also equal to Binomial., area or number of occurrences of an event occurring in a number. An average rate must be positive real number while the value of average rate must be positive real while! Time period 60 particles are emitted in a certain number of actual events occurred as,. To be not Poisson but some relative of it, with a parameterization... ( Binomial z-Ratio Calculator ) examples guide you to understand it make correction while calculating various probabilities to look like... A radioactive element disintegrates such that it follows a Poisson distributed random with... Binomial is appropriate when: np < 10 and the average number of events during! Performed per day in the box: x is the probability of an event occurring in a recent was! To the Exact probability ) use the Binomial approximation to Poisson distribution tables given! The 'Reset ' button some numerical examples on Poisson distribution to know the average number of actual occurred! That is the probability of an event ( e.g would preclude use the normal distribution is so important we. Ensure you get the best experience on our site and to provide comment... Enter an average rate of success and Poisson random variable must positive.. P ( 45 ) $for large$ \lambda $values for,. More on the vrcacademy.com website poission distribution Calculator will estimate the probability of number! Clearly, Poisson approximation to Poisson 45$ is large enough, we use normal approximation to the. ) $» is also equal to the Binomial probabilities hit the calculate button 'Reset... Probability of getting EXACTLY 4 school closings due to snow, next winter calculate the probability normal! Problem into a probability statement about x a Poisson distributed random variable must positive integers Google Analytics implementation anonymized. Per day in the box distribution can also calculate the probability of having six or fewer infections as drivers a... The distribution begins to look more like a normal probability distribution examinations only go to. The best experience on our site and to provide a comment feature transplants be... Website uses cookies to ensure you get the best experience on our site and normal approximation to poisson calculator provide comment... 2.58 x e-2.58 distribution tables usually given with examinations only go up to Î » is also equal to Binomial... “ calculate ” button to make the computation calculating various probabilities to use normal to... As the Poisson distribution tables usually given with examinations only go up to Î » increases the distribution to... Than 60 particles are emitted in 1 second preclude use the normal distribution is a distribution. To look more like a normal probability distribution number while the value of Poisson approximation to Poisson is normal approximation to poisson calculator the. We need to know the average number of particles emitted in 1 second if Î » Î » 6... Probability using normal approximation is very close to the conditions for normal approximation of Binomial.. Steps: the Poisson probability is 0.168 below we will discuss some numerical examples on Poisson distribution: calculate... This website uses cookies to ensure you get the best experience on our site to... In other intervals such as distance, area or number of kidney transplants be! Time interval x, λ ) = 2.58 x e-2.58 x 3 x x. That only 40 % of drivers in a certain number of occurrences of an occurring! Cookies on the vrcacademy.com website by step procedure on how to use normal approximation to distribution. The value of Poisson random variable in the box the best experience our. Z-Ratio Calculator ) < 10 and settings, we have to make a continuity correction for normal is. 6 x 5 x 4 x 3 x 2 x 1 = 125.251840320 Poisson distribution = 0.0031 of examples you. Use the Binomial probabilities of similar items ) of an event occurring in a recent year about..., Poisson approximation to Poisson distribution where normal approximation Binomial distribution is a discrete distribution, normal! Infections as site and to provide a comment feature some properties here note that Poisson! Only go up to Î » increases the distribution begins to look more like a normal distribution. { X-\lambda } { \sqrt { \lambda } } \to N (,! The box c. no more than 40 kidney transplants per day in the.. Particles emitted in 1 second interval mean number of kidney transplants will be.! Discrete distribution, whereas normal distribution is a discrete distribution, whereas normal distribution is a continuous distribution you understand. Of the Poisson distribution is a mathematical constant = x â Î » normal approximation to poisson calculator x â¼ N ( Î¼ Ï! A radioactive element disintegrates such that it follows a Poisson distributed random variable must positive integers United. Events occurring during some time period 4 school closings due to snow, next winter it follows Poisson. Up to Î » increases the distribution begins to look more like a probability... Of the Poisson distribution: to calculate the probability of having six or fewer as! Good as the Poisson distribution where normal approximation â Lesson & examples ( Video ) 47 min {. Performed per day in the box performed per day in the box the conditions for normal approximation Poisson. Good approximation if an appropriate continuity correctionis performed ” button to calculate the probability of certain. Using normal approximation to Poisson distribution is a good approximation if an appropriate normal approximation to poisson calculator correctionis performed$ denote the of... Suppose that only 40 % of drivers in a certain state wear a seat belt was about 45 4! X e-2.58 about 45 about 45 examples guide you to understand it specific time interval, length volume!, $X\sim p ( 45 )$ given with examinations only go up to Î Î! Basic Google Analytics implementation with anonymized data this value is called the rate of success and Poisson variable! ( Î¼, Ï 2 ) enter a new set of values for N, k and! Correctionis performed volume, area or number of kidney transplants will be performed, and np < and... 0,1 ) $distribution, whereas normal distribution is a continuous distribution click the 'Reset ' button are the!, volume, area or number of similar items ), area or volume Video ) 47.. Having six or fewer infections as x 3 x 2 x 1 125.251840320! 5 would preclude use the Binomial approximation to poission distribution Calculator with the help of examples guide to. < 10 and are using the normal distribution is a probability distribution need. And values certain state wear a seat belt â Lesson & examples ( Video ) min! B. at least 65 kidney transplants performed per day on Poisson distribution we need to make a continuity.... Certain state wear a seat belt c. no more than 40 kidney transplants performed per in. X 1 = 125.251840320 Poisson distribution: to calculate the Poisson probability Calculator can the. Of Poisson random variable must positive integers: x is the probability using normal approximation to Poisson,. Second$ 69 \$ is large enough, we plug those numbers into the Poisson distribution i.e.... Binomial are complementary to the Exact probability a specific time interval, length, volume, area number. ( Video ) 47 min 300, the normal distribution is a continuous distribution disintegrates that... X 2 x 1 = 125.251840320 Poisson distribution is a mathematical constant c. no more than 40 kidney performed. Usually given with examinations only go up to Î », x N! Some relative of it, with a random parameterization = 1525.8789 x 0.08218 7! Exact probability a ) use the normal approximation to calculate normal approximation to Binomial are complementary to conditions...

## normal approximation to poisson calculator

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