Question

1. Let X be a continuous random variable with CDF F(ro)-a+b 3 and support set 0, 1]. (a) Calculate the values of a, b that would make F(ro) a valid CDF. (b) Write out the pdf of X. c) Calculate EX d) Calculate EX

Answer 1

a) The CDF $F(x_0)=a+bx_0^2$ is defined in the interval [0,1].

To be a valid CDF, the function should satisfy

$\lim_{x_0\rightarrow-\infty}F(x_0)=0$

Since the support is in the interval [0,1], the value of $F(x_0)=0$ for all $x_0<0$

That means at $x_0=0$,  

$\begin{align*} &F(0)=0\\ \implies &a&plus;b\times 0=0\\ \implies &a=0 \end{align*}$

To be a valid CDF, the function should satisfy

$\lim_{x_0\rightarrow \infty}F(x_0)=1$

Since the support is in the interval [0,1], the value of $F(x_0)=1$ for all $x_0>1$

That means at $x_0=1$,  

$\begin{align*} &F(1)=1\\ \implies &a&plus;b\times 1=1\\ \implies &b=1-a\\ \implies &b=1\quad\text{since }a=0 \end{align*}$

For Valid CDf, a=0 and b=1 and the CDf is

$\begin{align*} F(x)=x^2,\quad 0\le x\le 1 \end{align*}$

b) The pdf is calculated using

$\begin{align*}f(x) &=\frac{d}{dx}F(x)\\ &=\frac{d}{dx}x^2\\ &=2x,\quad 0\le x\le 1 \end{align*}$

Formally the pdf is

$\begin{align*}f(x) &=\begin{cases} 2x,& 0\le x\le 1\\0,&\text{otherwise}\end{cases} \end{align*}$

c) The expected value of X is

$\begin{align*}E(X)&=\int_{-\infty}^{\infty}xf(x)dx \\ &=\int_0^1x(2x)dx\\ &=\int_0^12x^2dx\\ &=2\left(\left.\frac{x^3}{3}\right|_0^1 \right )\\ &=\frac{2}{3} \end{align*}$

d) The expected value of $\begin{align*}X^2 \end{align*}$ is

$\begin{align*}E(X^2)&=\int_{-\infty}^{\infty}x^2f(x)dx \\ &=\int_0^1x^2(2x)dx\\ &=\int_0^12x^3dx\\ &=2\left(\left.\frac{x^4}{4}\right|_0^1 \right )\\ &=\frac{1}{2} \end{align*}$

e) Variance of X ?

$\begin{align*}Var(X)=E(X^2)-[E(X)]^2=\frac{1}{2}-\left(\frac{2}{3} \right )^2=\frac{1}{18} \end{align*}$