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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 !!

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

(a) Find the probability distribution of s.

(b) Find the probability distribution of

Cumulative Distribution Function

Normal distribution

(a) We can write

Let Cumulative distribution function of S be denoted by . Then ,

----(1)

Here given that Y takes values 1 and -1 with equal probabilities .so , .

Now as hence X is symmetric distribution about 0 . Thus X and -X are identically distributed .

Thus from (1) we get =

Hence .

(b) Let K= =

Let C.D.F of K be

= as .

Find the distribution of T .

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 !!

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

(a) Find the probability distribution of s.

(b) Find the probability distribution of

Cumulative Distribution Function

Normal distribution

(a) We can write

Let Cumulative distribution function of S be denoted by . Then ,

----(1)

Here given that Y takes values 1 and -1 with equal probabilities .so , .

Now as hence X is symmetric distribution about 0 . Thus X and -X are identically distributed .

Thus from (1) we get =

Hence .

(b) Let K= =

Let C.D.F of K be

= as .

Find the distribution of T .

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