The number of people that enter a drugstore in a given hour is a Poisson random variable with parameter λ = 10. Compute the conditional prob- ability that at most 3 men entered the drugstore, given that 10 women entered in that hour. What assumptions have you made?

Answer:The probability is 0.2650Step-by-step explanation:Let's start assuming that men and women come in at the same rate.Let's define the following random variables : X : ''Number of people that enter a drugstore''M : ''Number of men that enter a drugstore''W : ''Number of women that enter a drugstore''The number of people will be the number of men plus the number of women⇒X = M + W We are also assuming that M and W are independent random variables.X ~ Po (10)M ~ Po (λ1)W ~ Po (λ2)λ1 = λ2 because we assumed that men and women come in at the same rate.λ1 = λ2 = λλ1 + λ2 = λ + λ ⇒ 2λ = 10 ⇒ λ = 5M ~ Po (5) W ~ Po (5)Because X is the sum of two independent Poisson random variables.We are looking for :[tex]P(M\leq 3/W=10)=P(M\leq 3)[/tex]Because we assume independence.[tex]P(M\leq 3)= P(M=0)+P(M=1)+P(M=2)+P(M=3)[/tex][tex]P(M=m)=\frac{5^{m}}{m!}e^{-5}[/tex] because is a Poisson random variable with λ = 5[tex]P(M\leq 3)=\frac{5^{0}}{0!}e^{-5}+\frac{5^{1}}{1!}e^{-5}+\frac{5^{2}}{2!}e^{-5}}+\frac{5^{3}}{3!}e^{-5}[/tex][tex]P(M\leq 3)=e^{-5}+5e^{-5} +(\frac{25}{2})e^{-5} +\frac{125}{6}e^{-5}=0.2650[/tex][tex]P(M\leq 3)=0.2650[/tex]