# Poisson Distribution Example for λ = 3 & x = 11

The below summary & example workout with step by step calculation shows how to estimate P(x), *mean* (μ), *variance* (σ^{2}) & *standard deviation* (σ) of Poisson *probability *distribution for the input values λ = 3, x = 11 may helpful for beginners to learn how the input parameters are being used in the Poisson formula in probability & statistical experiments or useful for grade school students to solve such similar worksheet or homework problems efficiently.

Calculation Summary | |
---|---|

Success Rate (λ) | 3 |

Random Variable (X) | 11 |

P(X = 11) | 0.0002 |

Mean (μ) | 3 |

Variance (σ^{2}) | 3 |

Standard Deviation (σ) | 1.7321 |

## Solved Example Calculation for λ = 3 & x = 11

This below solved example for Poisson probability distribution with step by step calculation shows how the input parameters λ = 3 & x = 11 are being used in the Poisson distribution formula to find the probability of success P(x).

__Work with step by step calculation :__

step 1 Address the formula, input parameters and values

Success Rate (λ) = 3

Random Variable (x) = 11

step 2 Substitute success rate & random variable in the below Poisson probability distribution formula

P(X = x) =e

^{-λ}λ

^{x}x!

P(X = 11) =e

^{-3}3

^{11}11!

step 3

*To find the factorial for 11*

11! = 1 x 2 x 3 x . . . . x 10 x 11

11! = 39916800

step 4 Simplify the above expression

P(X = 11) =e

^{-3}3

^{11}39916800

=0.0498 x 17714739916800

=8819.629839916800

= 0.0002

P(X = 11) = 0.0002

Mean (μ) = 3

Variance (σ

^{2}) = 3

Standard deviation (σ) = 1.7321

Thus, the probabilty of success P(x) is 0.0002 for λ = 3, x = 11.