Let U ~ U (0, 1). Let a and b be constants with a

a. Find the cumulative distribution function of U (use the result of Exercise 16).

b. Show that P(X≤ x) = P(U≤ (x-a)/(b - a)).

c. Use the result of part (b) to show that X ~ U (a, b).

Answer:

Step 1 of 4:

Given,

Let x be a continuous random variable with probability density function

f(x) = ,

We have U~U(0,1)

That is, f(x) = 1

Step 2 of 4:

The claim is to find the cumulative distribution function of UFrom the definition of cdf

F(x) = f(x) dx

If x<0, we have that f(x) =0, so F(x) = 0

If, 0 < x 1

F(x) = f(x) dx

Where, f(x) = ,

F(x) =dx

= (x

= x

Therefore, P(X x) = x, 0 < x 1

If, x = 1

F(x) = f(x) dx

= x

= 1

Hence, the cumulative distribution of x is