# Thread: pdf and cdf. mean and variance of the pdf

1. ## pdf and cdf. mean and variance of the pdf

Hello! there are two problems:

1) the random variable X has the following pdf : 1/4 if 0<x<4, and 0 otherwise.
determine the pdf and cdf for the random variable Y=X^1/2.
what is the expected value of Y?
what is the variance of Y?

2) calculate the mean and the variance of returns if they are modeled according to the following pdf: f(x)=1/4 if -1<x<3 , and otherwise f(x)=0.

2. Originally Posted by anna30
Hello! there are two problems:

1) the random variable X has the following pdf : 1/4 if 0<x<4, and 0 otherwise.
determine the pdf and cdf for the random variable Y=X^1/2.
what is the expected value of Y?
what is the variance of Y?

2) calculate the mean and the variance of returns if they are modeled according to the following pdf: f(x)=1/4 if -1<x<3 , and otherwise f(x)=0.

Note that $\displaystyle E(X^n) = \int_{-\infty}^{+\infty} x^n f(x) \, dx$ and $\displaystyle Var(X) = E(X^2) - (E(X))^2$. These are things you're meant to know, as well as being able to integrate.
3. Find the transformation $\displaystyle T(y) = \sqrt{x}$ Its inverse $\displaystyle T^{-1}(y)$ = y^2
$\displaystyle f_{Y}(y)=f_{X}(T^{-1}(y))|\frac{dT(y)}{dy}|$