probability - Continuous random variable on [0,1]

Let $X$ be a continuous random variable defined on the interval $[0, 1]$ with density function $f_X (u)=c(1-u^2)$ for a suitable constant $c$. Find $c$, and so find the expectation and the variance of $X$. Also find the density function of the random variable $Y = X^2$

I got $\int_{0}^{1}c(1-u^2)du=1$ so $c=\frac{3}{2}$.

$E(X)=\int_{0}^{1}\frac{3}{2}u(1-u^2) du=\frac{3}{8}$

I am stucked here and not sure if the results are right

Answer

Your calculation of $c$ is wrong. If you show how you calculated $c$, I can help you find the mistake.

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Your method for calculating $E(X)$ is correct, but your result is wrong because $c$ is not calculated correctly.

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To calculate the variance of $X$, you can use the formula available here

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To calculate the density of $Y$, it's easiest to calculate the cumulative distribution function of $Y$ first.