Q.6.24 - A First Course in Probability

Question

Consider independent trials, each of which results in outcome i, i = 0, 1, ... , k, with probability pi,  k i=0 pi = 1. Let N denote the number of trials needed to obtain an outcome that is not equal to 0, and let X be that outcome. 

(a) Find P{N = n}, n Ú 1. 

(b) Find P{X = j}, j = 1, ... , k. 

(c) Show that P{N = n, X = j} = P{N = n}P{X = j}. 

(d) Is it intuitive to you that N is independent of X? 

(e) Is it intuitive to you that X is independent of N? 

Step-by-Step Solution

Answer

a. $P (N=n)={p_0}^{n-1}. (1-p_0)$

b. $P [X=j]=\frac{p_j}{1-p_0}$

c. $P (X=j, N=n)=P(X=j).P (N=n),$ where X and N are independent.

d. Yes, N is independent of X.

e. Yes, X is independent of N.

Step 1: Content Introduction

Observe that random variables X and N are independent.

Step 2: Explanation (Part a)

We observe that N has geometric distribution with parameter of success 1 - p₀ .

Hence, $P (N=n)= {p_0}^{n-1}.(1-p_0)$.

This is because of the fact that N = n, it implies that in first n - 1 trials we have obtained outcomes equal to zero and in the nth trial we have obtained any other trial rather than zero.

Stepp 3: Explanation (Part b)

Whatever is the time of first non zero outcome, every other non-zero outcome has proportional probability to be that  outcome,

$P (X=j)=\frac{p_j}{1-p_0}$

Observe that we divide p_j with 1 - p₀ since that we know considered outcome cannot be zero and we want sum of all probabilities is equal to one.

Step 4: Explanation (part c)

We have that,

$P (X=j, N=n)= P(X=j/ N=n). P(N=n)$

Since the probability $P (X=j)$ is equal no matter which time of first non zero outcome is, we have that $P (X=j/ N=n)=P (X=j)$

Hence, $P (X=j, N=n)= P(X=j).P(N=n)$.

So, we have proved that X and N are independent.

Step 5: Explanation (Part d)

Yes, as its known what is the outcome of first non-zero outcome does not imply when the first non zero outcome might happen. Hence, N is independent of X. 

Step 6: Explanation (Part e)

Yes, as its know the time the first non zero outcome does not tell us what is that outcome. The probabilities remain the same. Hence, X is independent of N.