Question

a random variable X has density function f(x) = cx2 for 0<x<3 and f(x)= 0 others. a. Find constant value o b. Find probability P(1 < X < 2)

help

Answer 1

Given that a random variable X has density function,

fx(C) = cr2,0<x<3 0 otherwise ?

Now before we go on to solve the let us know two important properties of a Probability Density Function.

Two Important Properties of a Probability Density Function

$\bullet\emph{ } f_X(x)\geq 0, \emph{ for all x}$

$\bullet\emph{ }\int_{-\infty }^{\infty }f_{X}(x)dx=1$

Coming back to our problem

(a) Here we need to find the value of the constant c.

Here we will use the second property to find the value of c.

$\int_{-\infty }^{\infty }f_{X}(x)dx=1$

$\Rightarrow \int_{0 }^{3 }f_{X}(x)dx=1$

$\Rightarrow \int_{0 }^{3 }cx^{2}dx=1$

$\Rightarrow c\int_{0 }^{3 }x^{2}dx=1\emph{ }[\emph{Since c is a constant}]$

$\Rightarrow c\begin{bmatrix} \frac{x^{3}}{3} \end{bmatrix}_{0}^{3}=1$

$\Rightarrow c\begin{bmatrix} \frac{3^{3}}{3}-\frac{0^{3}}{3} \end{bmatrix}=1$

$\Rightarrow c\begin{bmatrix} 9-0 \end{bmatrix}=1$

$\Rightarrow 9c=1$

$\mathbf{\Rightarrow c=\frac{1}{9}}$

Hence the probability density function is,

$f_{X}(x)=\left\{\begin{matrix} \frac{x^{2}}{9} &,0<x<3 \\ 0&,otherwise \end{matrix}\right.$

(b) Here we need to find the probability P(1<X<2),

$=\int_{1}^{2}f_{X}(x)dx$

$=\int_{1}^{2}\frac{x^{2}}{9}dx$

$=\frac{1}{9}\int_{1}^{2}x^{2}dx$

$=\frac{1}{9}\begin{bmatrix} \frac{x^{3}}{3} \end{bmatrix}_{1}^{2}$

$=\frac{1}{9}\begin{bmatrix} \frac{2^{3}}{3}-\frac{1^{3}}{3} \end{bmatrix}$

$=\frac{1}{9}\begin{bmatrix} \frac{8}{3}-\frac{1}{3} \end{bmatrix}$

$=\frac{1}{9}*\frac{7}{3}$

$=\frac{7}{27}$

$\mathbf{\Rightarrow P(1<X<2)=\frac{7}{27}}$