Question

X is a continuous random variable with the cumulative distribution function

F(x) = 0 when x < 0

= x^{2}
when 0 ≤ x ≤
1

= 1 when x > 1

- Compute P(1/4 < X ≤ 1/2)

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

- 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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