Question

Suppose that X is continuous random variable with PDF f(x) and CDF F(x). (a) Prove that if f(x) > 0 only on a single (possibly infinite) interval of the real numbers then F(x) is a strictly increasing function of x over that interval. [Hint: Try proof by contradiction]. (b) Under the conditions described in part (a), find and identify the distribution of Y = F(x).

Answer #1

Given pdf and CDF . Now the property of pdf is

.

Now

That is is a strictly increasing function of over the interval .

Given the transformation .

The pdf of is

The distribution of uniform in the interval

Suppose a random variable X has cumulative distribution function
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Note: Here, Y is...

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f(x) =
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x/4 if 0 < or = x< 2
1/2 if 2 < or = x< 3
0 if x> or = 3
(a) Find P(X<1)
(b) Find P(X<2.5)
(c) Find the cumulative distribution function F(x) = P(X< or
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