Question

X is a continuous random variable with the cumulative distribution function

F(x)   = 0               when x < 0

= x²              when 0 ≤ x ≤ 1

= 1               when x > 1

1. Compute P(1/4 < X ≤ 1/2)

2. What is f(x), the probability density function of X?

3. Compute E[X]

Answer 1

We are given the cumulative distribution function of X:

Part 1

The required probability is given by:

Part 2

The probability density function of X is given by:

Thus, the pdf of X is given by:

Part 3

The expected value of X is given by:

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