Question

A random variable X has the probability density function given below

f(x)=$\begin{Bmatrix} x, 0<=x<1 \\ 2-x, 1<=x<2\\ 0, otherwise \end{Bmatrix}$ Show graphically and find out the robabilities. p(-1<x<55)

Answer 1

$$f(x) = \begin{cases} 0 &x\leq 0\\ x &0\leq x< 1 \\ 2-x &1\leq x< 2 \\ 0 &x\geq 2 \\ \end{cases}$$

$$\displaystyle\int_{-\infin}^{\infin}f(y)dy=\dfrac{1}{2}(2-0)(1)=1$$

$$F(x)=\displaystyle\int_{-\infin}^{x}f(y)dy$$

$$P(a\leq X\leq b)=F(b)-F(a)$$

Then

$$P(-1<X<55)=1$$