2022-01-30

What are the mean and standard deviation of a probability density function given by $Pr\left(X=k\right)=\frac{{24}^{k}{e}^{-24}}{k!}$ for $k\in \left\{0,1,2...\mathrm{\infty }\right\}$?

gekraamdbk

Expert

Explanation: $P\left(X=k\right)=\frac{{24}^{k}{e}^{-24}}{k!}$ for $k\in \left\{0,1,2...\mathrm{\infty }\right\}$ is a Poisson distribution which is of teh form $P\left(X=k\right)=\frac{{\lambda }^{k}{e}^{-\lambda }}{k!}$ for $k\in \left\{0,1,2...\mathrm{\infty }\right\}$ This means that $E\left(X\right)=\lambda$ and $Var\left(X\right)=\lambda$ for the given distribution E(X)=24 Var(X)=24 $sd=\sqrt{Var\left(X\right)}=\sqrt{24}=2\sqrt{6}$