Question

Let N = the number of the tossing a coin when we get the first HEAD. M = the number of the tossing a coin when we get the second HEAD. We assume probability of getting a HEAD is p. Find the Probability Distribution of N and M and are they independent?

Answer 1

The random variable N is the number of toss until first head

N follow Geometric distribution or Negative Binomial distribution with probability of success = p and r=1 (number of success)

The probability mass function of N is

P(N=n, r=1) = (1-p)n-1 p, n=1, 2,.....

The random variable M is the number of toss until second head

M follow Negative Binomial distribution with probability of success = p , r= 2 (number of success)

The probability mass function of M is

P(M=m, r=2) =m-1Cm-2 (1-p)m-2 p2 = (m-1)(1-p)m-2 p2 , m=2,3,....

as

The sample Sapce of N and M are given below

SN= {H,TH,TTH, TTTH,....}

SM={HH,THH,TTHH,THTH,TTTHH,.....}

M and N are not independent . P( M=m I N=n) is the probability of number of tosses to get another head , given that one head has already occurred. Thus M is not independent of N .