Question

4. Let X be the number of distinct birthdays in a group of 100 people. In a webwork, you found the expected value of x. You should now find its variance (assuming each person was born on any of 365 days equally likely, and independently from any other person).

Answer 1

4.

Xi = 1 if atleast one person has birthday on i th day , 0 otherwise

P(Xi = 1) = 1 - P(Xi = 0)

= 1 - (p(particular person doesn't have birthday on that day))^100  

{p(particular person doesn't have birthday on that day) = 364/365}

= 1 - (364/365)^100

= 0.2399

X : no. of days with Xi = 1

binomial distribution with p = 0.2399

E(X) in binomial distribution = n*p

E(X) = 100*0.2399 = 23.99

E(X) = 23.99

Var(X) in binomial distribution = n*p*(1-p)

= 100*0.2399*(1-0.2399)

Var(X) = 18.2348

(please UPVOTE)