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Consider for $c\in \mathbb R $ the function $f:\mathbb R→\mathbb R$ is defined by

$$f(t)=\begin{cases} c(t-1), & 1 \leq t <2\\ c, & 2 \leq t <5 \\ \frac{-c}{2}(t-7), & 5 \leq t <7 \\ 0, & \text{otherwise} \end{cases} $$

By choosing the appropriate $c$, $f$ is a density function. Let $X$ a random variable, it's density function is $f$.

Consider the random variable $Y=X^2+2$. How to calculate correlation $ρ_{XY}$? Decide whether the random variables $X$ and $Y$ are uncorrelated or independent.

$c=\frac{2}{9}$

I wanted to use this formula:

$corr(X,Y)=\frac{cov(X,Y)}{\sigma_x \sigma_y}$

But what are $\sigma_x$ and $\sigma_y$ in my example?

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    The title could stand to be more descriptive of the problem, to help future searchers. – 2017-01-06

1 Answers 1

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Hint:

$$\sigma_x^2= \mathbb{E}[X^2]-\mathbb{E}[X]^2$$

$$\sigma_Y^2=Var(X^2+2)=Var(X^2)=\mathbb{E}[X^4]-\mathbb{E}[X^2]^2$$