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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 do I calculate the covariance $\text{Cov}(X,Y)$ and correlation coefficients $ρ_{XY}$? Decide whether the random variables $X$ and $Y$ are uncorrelated or independent.

I already calculated $c=\frac{2}{9}$ which was previous part of this example, but I dont know how to calculate covariance and correlation. $\text{Cov}(X,Y)=\text{Cov}(X,X^2+2)$ but what after this?

Winther
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Ana Matijanovic
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  • What does $f(t)$ represent? – Mark Viola Dec 30 '16 at 15:34
  • What is $f(t)$? – Canardini Dec 30 '16 at 15:35
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    You have the PDF of $X$ hence you can compute $E(X)$, $E(X^2)$ and $E(X^3)$, and then everything follows, for example the covariance of $X$ and $Y$ is $E(X(X^2+2))-E(X)E(X^2+2)=E(X^3)-E(X)E(X^2)$. – Did Dec 30 '16 at 15:37
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    @Dr.MV In full generality, $\mathrm{Cov}(X,Y)=E((X-E(X))(Y-E(Y)))=E(XY)-E(X)E(Y)$. – Did Dec 30 '16 at 15:43
  • @Did Thanks. Can you help me with correlation? I found definition $ρ_{XY}=\frac{Cov(X,Y)}{\sigma_x \sigma_y}$. I am not sure what is here $\sigma$. Is this correct: $\sigma_x=\sqrt{E\left[X^2\right]-(E\left[X\right] )^2 } ,

    \sigma_y=\sqrt{E\left[X^2+2\right]-(E\left[X^2+2\right] )^2 } $

     – Ana Matijanovic Dec 30 '16 at 16:12
  • Are you coming here to get definitions or to be helped solving exercises? 'Cause the site is done for the latter, not the former. – Did Dec 30 '16 at 16:54

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