Q. 5.7 - A First Course in Probability

Question

The density function of $X$ is given by

$f\left(x\right)=\left\{\begin{array}{l}a+b{x}^2; 0\le x\le 1\\ 0; \text{Otherwise}\end{array}\right.$

If  $E\left(X\right)=\frac{3}{5}$, find $a \text{and} b$.

Step-by-Step Solution

Answer

The value of $a=\frac{3}{5}$ and $b=\frac{6}{5}$.

Step 1. Given information.

It is given that the density function of $X$ is

$f\left(x\right)=\left\{\begin{array}{l}a+b{x}^2; 0\le x\le 1\\ 0; \text{Otherwise}\end{array}\right.$

and

$E\left(X\right)=\frac{3}{5}$

Step 2. Solve the given density function.

The given density function is 

$f\left(x\right)=\left\{\begin{array}{l}a+b{x}^2; 0\le x\le 1\\ 0; \text{Otherwise}\end{array}\right.$

According to the given information, the area under the density curve is $1$.

So,

$$\begin{gathered} {\int }_{-\infty }^{\infty }f\left(x\right) dx = 1 \\ \Rightarrow {\int }_0^1\left(a+b{x}^2\right) dx = 1 \\ \Rightarrow {\left[ax+\frac{b{x}^3}{3}\right]}_0^1=1 \\ \Rightarrow a\left(1-0\right)+\frac{b\left(1^3-0^3\right)}{3}=1 \\ \Rightarrow a+\frac{b}{3}=1 \\ \Rightarrow 3a+b=3 .................. \left(1\right) \end{gathered}$$

Step 3. Solve the given density function for the expected value of the random variable.

It is given that $E\left[X\right]=\frac{3}{5}$.

According to the given information, the expectation for a continuous random variable is $E\left[X\right]={\int }_{-\infty }^{\infty }f\left(x\right) dx =1$

$$\begin{gathered} \Rightarrow \frac{3}{5}={\int }_0^{1}x\left(a+b{x}^2\right) dx \\ \Rightarrow \frac{3}{5}={\int }_0^1\left(ax+b{x}^3\right) dx \\ \Rightarrow \frac{3}{5}={\left[\frac{a{x}^2}{2}+\frac{b{x}^4}{4}\right]}_0^1 \\ \Rightarrow \frac{3}{5}=\left[\frac{a\left(1^2-0^2\right)}{2}+\frac{b\left(1^4-0^4\right)}{4}\right] \\ \Rightarrow \frac{3}{5}=\frac{a}{2}+\frac{b}{4} \\ \Rightarrow 12=10a+5b ...................... \left(2\right) \end{gathered}$$

Step 4. Find the value of a   and   b .

Multiplying equation $\left(1\right)$ by $5$ we get,

$15a+5b=15$

Subtracting equation $\left(2\right)$ from $\left(3\right)$ we get,

$$\begin{gathered} 5a=3 \\ \Rightarrow a=\frac{3}{5} \end{gathered}$$

Substituting the value of $a$ in equation $\left(1\right)$ we get,

$$\begin{gathered} 3\left(\frac{3}{5}\right)+b=3 \\ \Rightarrow b=3-\frac{9}{5} \\ \Rightarrow b=\frac{15-9}{5}=\frac{6}{5} \end{gathered}$$

Therefore, the value of $a=\frac{3}{5} \text{and} b=\frac{6}{5}$ and

$$\begin{gathered} f\left(x\right) = \frac{3}{5}+\frac{6}{5}{x}^2 \\ =\frac{3}{5}\left(1+2x^2\right) \end{gathered}$$

So, the density function of $X$ will be

$f\left(x\right)=\left\{\begin{array}{l}\frac{3}{5}\left(1+2x^2\right), 0\le x\le 1\\ 0, \text{Otherwise}\end{array}\right.$