The Poisson Distribution Calculator will construct a complete poisson distribution, and identify the mean and standard deviation.  A poisson probability is the chance of an event occurring in a given time interval.  Enter $\lambda$ and the maximum occurrences, then the calculator will find all the poisson probabilities from 0 to max.  If you want to find the poisson probability for a specific occurrence only, then have a look at the Poisson Probability Calculator. Enter a value for $\lambda$ and x.  Lambda, $\lambda$, is the average number of occurrences for a given time interval.  x is the maximum number of occurrences.  The calculator will display the poisson distribution, the mean, and the standard deviation.  Then, it will also give you a step by step solution for how to find poisson probabilities.

Mean of the Poisson Distribution, $\mu$: 5.1
Standard Deviation of the Poisson Distribution, $\sigma$: 2.2583179581272

P(0) = 0.0060967465655156
P(1) = 0.03109340748413
P(2) = 0.079288189084531
P(3) = 0.1347899214437
P(4) = 0.17185714984072
P(5) = 0.17529429283754
P(6) = 0.1490001489119
P(7) = 0.1085572513501

### Solution:

The mean of the poisson distribution is interpreted as the mean number of occurrences for the distribution. By definition, $\lambda$ is the mean number of successes for a poisson distribution. For this distribution, the mean is $$\mu = \lambda = 5.1$$

The standard deviation of the poisson distribution is interpreted as the standard deviation of the number of occurences for the distribution. To find the standard deviation, use the formula $$\sigma = \sqrt{\lambda}$$ Substituting in the value of $\lambda$ for this problem, we have $$\sigma = \sqrt{5.1}$$ Evaluating the expression on the right, we have $$\sigma = 2.2583179581272$$

To complete a poisson distribution table, first identify all of the possible values of X. Since the maximum number of occurences is 7, the values of X range from X = 0 to X = 7.

Next, find each individual poisson probability for each value of X. In this problem, we will be finding 8 probabilities. The sum of all these probabilities will be 1.

### $P(0)$ Probability of exactly 0 occurrences

If using a calculator, you can enter $\lambda = 5.1$ and $x = 0$ into a poisson probability distribution function (poissonPDF). If doing this by hand, apply the poisson probability formula: $$P(x) = \frac{{e^{-\lambda}} \cdot {\lambda^x}}{x!}$$ where $x$ is the number of occurrences, $\lambda$ is the mean number of occurrences, and $e$ is the constant 2.718. Substituting in values for this problem, $x = 0$ and $\lambda = 5.1$, we have $$P(0) = \frac{{e^{-5.1}} \cdot {5.1^0}}{0!}$$ Remember, 0! is 1. Evaluating the expression, we have $$P(0) = 0.0060967465655156$$

### $P(1)$ Probability of exactly 1 occurrences

If using a calculator, you can enter $\lambda = 5.1$ and $x = 1$ into a poisson probability distribution function (poissonPDF). If doing this by hand, apply the poisson probability formula: $$P(x) = \frac{{e^{-\lambda}} \cdot {\lambda^x}}{x!}$$ where $x$ is the number of occurrences, $\lambda$ is the mean number of occurrences, and $e$ is the constant 2.718. Substituting in values for this problem, $x = 1$ and $\lambda = 5.1$, we have $$P(1) = \frac{{e^{-5.1}} \cdot {5.1^1}}{1!}$$ Evaluating the expression, we have $$P(1) = 0.03109340748413$$

### $P(2)$ Probability of exactly 2 occurrences

If using a calculator, you can enter $\lambda = 5.1$ and $x = 2$ into a poisson probability distribution function (poissonPDF). If doing this by hand, apply the poisson probability formula: $$P(x) = \frac{{e^{-\lambda}} \cdot {\lambda^x}}{x!}$$ where $x$ is the number of occurrences, $\lambda$ is the mean number of occurrences, and $e$ is the constant 2.718. Substituting in values for this problem, $x = 2$ and $\lambda = 5.1$, we have $$P(2) = \frac{{e^{-5.1}} \cdot {5.1^2}}{2!}$$ Evaluating the expression, we have $$P(2) = 0.079288189084531$$

### $P(3)$ Probability of exactly 3 occurrences

If using a calculator, you can enter $\lambda = 5.1$ and $x = 3$ into a poisson probability distribution function (poissonPDF). If doing this by hand, apply the poisson probability formula: $$P(x) = \frac{{e^{-\lambda}} \cdot {\lambda^x}}{x!}$$ where $x$ is the number of occurrences, $\lambda$ is the mean number of occurrences, and $e$ is the constant 2.718. Substituting in values for this problem, $x = 3$ and $\lambda = 5.1$, we have $$P(3) = \frac{{e^{-5.1}} \cdot {5.1^3}}{3!}$$ Evaluating the expression, we have $$P(3) = 0.1347899214437$$

### $P(4)$ Probability of exactly 4 occurrences

If using a calculator, you can enter $\lambda = 5.1$ and $x = 4$ into a poisson probability distribution function (poissonPDF). If doing this by hand, apply the poisson probability formula: $$P(x) = \frac{{e^{-\lambda}} \cdot {\lambda^x}}{x!}$$ where $x$ is the number of occurrences, $\lambda$ is the mean number of occurrences, and $e$ is the constant 2.718. Substituting in values for this problem, $x = 4$ and $\lambda = 5.1$, we have $$P(4) = \frac{{e^{-5.1}} \cdot {5.1^4}}{4!}$$ Evaluating the expression, we have $$P(4) = 0.17185714984072$$

### $P(5)$ Probability of exactly 5 occurrences

If using a calculator, you can enter $\lambda = 5.1$ and $x = 5$ into a poisson probability distribution function (poissonPDF). If doing this by hand, apply the poisson probability formula: $$P(x) = \frac{{e^{-\lambda}} \cdot {\lambda^x}}{x!}$$ where $x$ is the number of occurrences, $\lambda$ is the mean number of occurrences, and $e$ is the constant 2.718. Substituting in values for this problem, $x = 5$ and $\lambda = 5.1$, we have $$P(5) = \frac{{e^{-5.1}} \cdot {5.1^5}}{5!}$$ Evaluating the expression, we have $$P(5) = 0.17529429283754$$

### $P(6)$ Probability of exactly 6 occurrences

If using a calculator, you can enter $\lambda = 5.1$ and $x = 6$ into a poisson probability distribution function (poissonPDF). If doing this by hand, apply the poisson probability formula: $$P(x) = \frac{{e^{-\lambda}} \cdot {\lambda^x}}{x!}$$ where $x$ is the number of occurrences, $\lambda$ is the mean number of occurrences, and $e$ is the constant 2.718. Substituting in values for this problem, $x = 6$ and $\lambda = 5.1$, we have $$P(6) = \frac{{e^{-5.1}} \cdot {5.1^6}}{6!}$$ Evaluating the expression, we have $$P(6) = 0.1490001489119$$

### $P(7)$ Probability of exactly 7 occurrences

If using a calculator, you can enter $\lambda = 5.1$ and $x = 7$ into a poisson probability distribution function (poissonPDF). If doing this by hand, apply the poisson probability formula: $$P(x) = \frac{{e^{-\lambda}} \cdot {\lambda^x}}{x!}$$ where $x$ is the number of occurrences, $\lambda$ is the mean number of occurrences, and $e$ is the constant 2.718. Substituting in values for this problem, $x = 7$ and $\lambda = 5.1$, we have $$P(7) = \frac{{e^{-5.1}} \cdot {5.1^7}}{7!}$$ Evaluating the expression, we have $$P(7) = 0.1085572513501$$