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, $\mu$: 6.9
Standard Deviation of the Poisson Distribution, $\sigma$: 2.6267851073127

P(0) = 0.0010077854290485
P(1) = 0.0069537194604347
P(2) = 0.0239903321385
P(3) = 0.05517776391855
P(4) = 0.095181642759498
P(5) = 0.13135066700811
P(6) = 0.15105326705932
P(7) = 0.14889536324419
P(8) = 0.12842225079811

### 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 $$\mu = \lambda = 6.9$$

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 $$\sigma = \sqrt{6.9}$$ Evaluating the expression on the right, we have $$\sigma = 2.6267851073127$$

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 $\lambda = 6.9$ 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 $\lambda = 6.9$, we have $$P(0) = \frac{{e^{-6.9}} \cdot {6.9^0}}{0!}$$ Remember, 0! is 1. Evaluating the expression, we have $$P(0) = 0.0010077854290485$$

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

If using a calculator, you can enter $\lambda = 6.9$ 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 $\lambda = 6.9$, we have $$P(1) = \frac{{e^{-6.9}} \cdot {6.9^1}}{1!}$$ Evaluating the expression, we have $$P(1) = 0.0069537194604347$$

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

If using a calculator, you can enter $\lambda = 6.9$ 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 $\lambda = 6.9$, we have $$P(2) = \frac{{e^{-6.9}} \cdot {6.9^2}}{2!}$$ Evaluating the expression, we have $$P(2) = 0.0239903321385$$

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

If using a calculator, you can enter $\lambda = 6.9$ 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 $\lambda = 6.9$, we have $$P(3) = \frac{{e^{-6.9}} \cdot {6.9^3}}{3!}$$ Evaluating the expression, we have $$P(3) = 0.05517776391855$$

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

If using a calculator, you can enter $\lambda = 6.9$ 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 $\lambda = 6.9$, we have $$P(4) = \frac{{e^{-6.9}} \cdot {6.9^4}}{4!}$$ Evaluating the expression, we have $$P(4) = 0.095181642759498$$

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

If using a calculator, you can enter $\lambda = 6.9$ 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 $\lambda = 6.9$, we have $$P(5) = \frac{{e^{-6.9}} \cdot {6.9^5}}{5!}$$ Evaluating the expression, we have $$P(5) = 0.13135066700811$$

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

If using a calculator, you can enter $\lambda = 6.9$ 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 $\lambda = 6.9$, we have $$P(6) = \frac{{e^{-6.9}} \cdot {6.9^6}}{6!}$$ Evaluating the expression, we have $$P(6) = 0.15105326705932$$

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

If using a calculator, you can enter $\lambda = 6.9$ 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 $\lambda = 6.9$, we have $$P(7) = \frac{{e^{-6.9}} \cdot {6.9^7}}{7!}$$ Evaluating the expression, we have $$P(7) = 0.14889536324419$$

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

If using a calculator, you can enter $\lambda = 6.9$ 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 $\lambda = 6.9$, we have $$P(8) = \frac{{e^{-6.9}} \cdot {6.9^8}}{8!}$$ Evaluating the expression, we have $$P(8) = 0.12842225079811$$