The random variable X

Question

The random variable X has density function $$f(x)=(K+1)x^2 $$

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$K$ is constant. What is the value of $K$? and find the expected value of $E(x)$.

Answer 1

Set

$$\int\limits_{x=0}^1 (k+1) x^2\ dx = 1$$

to find $k = 2$.

Then $${\cal E}[x] = \int\limits_{x=0}^1 x\ 3 x^2\ dx = \frac{3}{4}$$