Let Z ~ N(0, 1), and let X = σZ +µ where p and σ > 0 are constants. Let Φ represent the cumulative distribution function of Z, and let Φ represent the probability density function, so

a. Show that the cumulative distribution function of

b. Differentiate Fx(x) to show that X ~ N(µ, σ2).

c. Now let X = -σZ +µ. Compute the cumulative distribution function of X in terms of Φ, then differentiate it to show that X ~ N(µ, σ2).

Answer:

Step 1 of 3:

(a)

In this question, we are asked to prove the cumulative distribution function of

Let , and let and is constants.

represent cumulative distribution function of , let represent the probability density function, so

We can write cumulative distribution function of random variable as

……….(1)

Given

Put the value of into the equation (1), this method is also called change a variable method

………..(2)

We can write cumulative distribution function of random variable as

……….(3)

equate equation (2) and (3), we can deduce the value of

Put the value of z into the equation (3), we will get,

………….(4)

From equation (2) and (4),we get

=

Hence proved.