Question

A coin is tossed with P(heads) = p.

a) What is the expected number of tosses required to get n heads?

b) Determine the variance of the number of tosses needed to get the first head.

c) Determine the variance of the number of tosses needed to get n heads.

Answer 1

X: Number of trails to get 'n' successes;

p : Probability of success;

Then X follows a Negative Binomial Distribution;

Expected value of X i.e Expected number of trails required to get 'n' successes :

Variance of X i.e Variance of the number of trails required to get 'n' successes :

For the given problem ,

Success : Getting a head

Trail : tossing of a coin

X : Number of tosses required to get 'n' heads

Therefore 'X' follows negative binomial distribution with 'p' : probability of success = P(heads) = p

a) What is the expected number of tosses required to get n heads

the expected number of tosses required to get n heads =

b) Determine the variance of the number of tosses needed to get the first head

for this case n=1

variance of the number of tosses needed to get the first head

c) Determine the variance of the number of tosses needed to get n heads

the variance of the number of tosses needed to get n heads