Question

A police department reports that the probabilities that 0, 1, 2, and 3 burglaries will be reported in a given day are 0.53, 0.40, 0.05, and 0.02, respectively. find the standard deviation for the probability distribution. round answer to the nearest hundredth.

Answer 1

The mean of the probability distribution is given by the expected value given by

$E(x)=\sum xp(x) \\ \\ =0(0.53)+1(0.40)+2(0.05)+3(0.02) \\ \\ =0.40+0.10+0.06=0.56$

The variance of the probability distribution is given by

$Var(x)=(x-\bar{x})^2p(x) \\ \\ =(0-0.56)^2(0.53)+(1-0.56)^2(0.40)+(2-0.56)^2(0.05)\\+(3-0.56)^2(0.02) \\ \\ =(-0.56)^2(0.53)+0.44^2(0.40)+1.44^2(0.05)+2.44^2(0.02) \\ \\ =0.3136(0.53)+0.1936(0.40)+2.0736(0.05)+5.9536(0.02) \\ \\ =0.166208+0.07744+0.10368+0.119072=0.4664$

The standard deviation is given by

$\sigma(x)=\sqrt{Var(x)} \\ \\ =\sqrt{0.4664}=0.683$