This is a very beautiful sample problem from ISI MStat PSB 2004 Problem 7 based on finding the distribution of a random variable . Let’s give it a try !!

Problem– ISI MStat PSB 2004 Problem 7


Suppose X has a normal distribution with mean 0 and variance 25 . Let Y be an independent random variable taking values -1 and 1 with
equal probability. Define \( S=X Y+\frac{X}{Y} \) and \( T=X Y-\frac{X}{Y}\)
(a) Find the probability distribution of s.
(b) Find the probability distribution of \( (\frac{S+T}{10})^{2}\)

Prerequisites


Cumulative Distribution Function

Normal distribution

Solution :

(a) We can write \( S = \begin{cases} 2x & , if Y=1 \\ -2x & , if Y=-1 \end{cases} \)

Let Cumulative distribution function of S be denoted by \( F_{S}(s) \) . Then ,

\( F_{S}(s) = P(S \le s) = P(S \le s | Y=1)P(Y=1) + P(S \le s| Y=-1)P(Y=-1) = P(2X \le s) \times \frac{1}{2} + P(-2X \le s) \times \frac{1}{2} \) —-(1)

Here given that Y takes values 1 and -1 with equal probabilities .so , \( P(Y=1)=P(Y=-1)= \frac{1}{2} \) .

Now as \(X \sim N(0, 5^2)\) hence X is symmetric distribution about 0 . Thus X and -X are identically distributed .

Thus from (1) we get \( F_{S}(s) = P(X \le s/2 ) \times \frac{1}{2} + P(-X \le s/2) \times \frac{1}{2} = P(X \le s/2 ) \times \frac{1}{2} + P(X \le s/2) \times \frac{1}{2}\)=\( P( X \le s/2) = P(\frac{X-0}{5} \le s/2 ) = \Phi(\frac{s-0}{10}) \)

Hence \( S \sim N(0,{10}^2) \).

(b) Let K=\( (\frac{S+T}{10})^{2}\) = \( \frac{{XY}^2}{ {(10)}^2} \)

Let C.D.F of K be \( F_{K}(k) = P(K \le k ) = P(K \le k | Y=1)P(Y=1) + P(K \le k| Y=-1)P(Y=-1) = P( \frac{X^2}{{(10)}^2} \le k ) \)

=\( P( -10 \sqrt{k} \le X \le 10 \sqrt{k} ) = \Phi(2\sqrt{k}) – \Phi(-2 \sqrt{k}) \) as \(X \sim N(0, 5^2)\).


Food For Thought

Find the distribution of T .


ISI MStat PSB 2008 Problem 10
Outstanding Statistics Program with Applications

Outstanding Statistics Program with Applications

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