Question

For the random variable X with the probability function below, answer the following questions.

f(x)= { 3x^2 , 0<x<1

{ 0, otherwise

a. Find the cdf of X.

b. Find the probability that X is less than 0.5.

c. Find E(X).

Answer 1

The probability function $f(x)=\begin{cases} 3x^2,\ \ \ 0<x<1 \\0, \ \ \text{otherwise} \end{cases}$

a. The CDF of X: $F(x)=P(X\leq x)=\int\limits _{-\infin}^x f(t)dt$

If $x\leq 0:$ $F(x)= \int\limits _{-\infin}^x 0dt =0$

If $0<x<1$ : $F(x)= \int\limits _{0}^x 3t^2dt =t^3\big|^x_0=x^3$

If $1\leq x$ : $F(x)= \int\limits _{0}^1 3t^2dt =t^3\big|^1_0=1$

$F(x)=\begin{cases} 0,\ \ x\leq 0 \\ x^3,\ \ 0<x<1 \\ 1,\ \ 1\leq x \end{cases}$

b. $P(X<0.5)= F(0.5)=(0.5)^3=0.125$

c. $E(x)=\int\limits _{-\infin}^{+\infin}xf(x)dx=\int\limits _0^1 3x^3dx=\frac{3}{4}x^4\big|^1_0=\frac{3}{4}$

Answer:

a. $F(x)=\begin{cases} 0,\ \ x\leq 0 \\ x^3,\ \ 0<x<1 \\ 1,\ \ 1\leq x \end{cases}$

b. $P(X<0.5)=0.125$

c. $E(x)=\frac{3}{4}$