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.

### Answer:

Mean of the Poisson Distribution, $\mu$: 6.5

Standard Deviation of the Poisson Distribution, $\sigma$: 2.5495097567964

P(0) = 0.0015034391929776

P(1) = 0.0097723547543542

P(2) = 0.031760152951651

P(3) = 0.068813664728578

P(4) = 0.11182220518394

P(5) = 0.14536886673912

P(6) = 0.15748293896738

P(7) = 0.14623415761257

P(8) = 0.11881525306021

### 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 $$ \mu = \lambda = 6.5 $$

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 $$ \sigma = \sqrt{6.5} $$ Evaluating the expression on the right, we have $$ \sigma = 2.5495097567964 $$

To complete a poisson distribution table, first identify all of the possible values of X. Since the maximum number of occurences is 8, the values of X range from X = 0 to X = 8.

Next, find each individual poisson probability for each value of X. In this problem, we will be finding 9 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 = 6.5 $ 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 = 6.5 $, we have
$$ P(0) = \frac{{e^{-6.5}} \cdot {6.5^0}}{0!} $$
Remember, 0! is 1.
Evaluating the expression, we have
$$ P(0) = 0.0015034391929776 $$

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

If using a calculator, you can enter $ \lambda = 6.5 $ 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 = 6.5 $, we have
$$ P(1) = \frac{{e^{-6.5}} \cdot {6.5^1}}{1!} $$
Evaluating the expression, we have
$$ P(1) = 0.0097723547543542 $$

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

If using a calculator, you can enter $ \lambda = 6.5 $ 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 = 6.5 $, we have
$$ P(2) = \frac{{e^{-6.5}} \cdot {6.5^2}}{2!} $$
Evaluating the expression, we have
$$ P(2) = 0.031760152951651 $$

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

If using a calculator, you can enter $ \lambda = 6.5 $ 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 = 6.5 $, we have
$$ P(3) = \frac{{e^{-6.5}} \cdot {6.5^3}}{3!} $$
Evaluating the expression, we have
$$ P(3) = 0.068813664728578 $$

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

If using a calculator, you can enter $ \lambda = 6.5 $ 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 = 6.5 $, we have
$$ P(4) = \frac{{e^{-6.5}} \cdot {6.5^4}}{4!} $$
Evaluating the expression, we have
$$ P(4) = 0.11182220518394 $$

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

If using a calculator, you can enter $ \lambda = 6.5 $ 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 = 6.5 $, we have
$$ P(5) = \frac{{e^{-6.5}} \cdot {6.5^5}}{5!} $$
Evaluating the expression, we have
$$ P(5) = 0.14536886673912 $$

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

If using a calculator, you can enter $ \lambda = 6.5 $ 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 = 6.5 $, we have
$$ P(6) = \frac{{e^{-6.5}} \cdot {6.5^6}}{6!} $$
Evaluating the expression, we have
$$ P(6) = 0.15748293896738 $$

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

If using a calculator, you can enter $ \lambda = 6.5 $ 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 = 6.5 $, we have
$$ P(7) = \frac{{e^{-6.5}} \cdot {6.5^7}}{7!} $$
Evaluating the expression, we have
$$ P(7) = 0.14623415761257 $$

### $P(8)$ Probability of exactly 8 occurrences

If using a calculator, you can enter $ \lambda = 6.5 $ and $ x = 8 $ 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 = 8 $ and $ \lambda = 6.5 $, we have
$$ P(8) = \frac{{e^{-6.5}} \cdot {6.5^8}}{8!} $$
Evaluating the expression, we have
$$ P(8) = 0.11881525306021 $$