Question

A random variable X has the probability density function given below

f (x)= {x ,0<=x< 1

{2-x, 1<=x<2 Show graphically and find out the robabilities. p(-1<x<a)

{ 0, otherwise

.

Answer 1

Solution:

$f(x)=\{x, \ \ \ \ \ 0\le x<1 \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2-x, 1\le x<2 \\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0, \ \ \ \ \ \ otherwise$

$P(-1< x<a)=P(-1<x<0)+P(0\le x<a)=0+P(0\le x<a) \\=P(0\le x<a)$

Now, we will have 2 cases:

Case I: When $a<1$

$P(0\le x<a)=\int_0^ax dx=(x^2/2)_0^a=0.5(a^2-0)=0.5a^2$

Case II: When $1\le a<2$

$P(0\le x<a)=P(0\le x<1)+P(1\le x<a) \\=\int_0^1x dx+\int_1^a(2-x) dx=(x^2/2)_0^1+(2x-x^2/2)_1^a \\=0.5(1-0)+2(a-1)-0.5(a^2-1)=0.5+2a-2-0.5a^2+0.5 \\=2a-0.5a^2-1$