Q.6.21 - A First Course in Probability

Question

Let $f(x, y) = 24xy 0 \le x \le 1, 0 \le y \le 1, 0 \le x + y \le 1$ and let it equal 0 otherwise.

(a) Show that $f(x, y)$ is a joint probability density function.

(b) Find $E\left[X\right]$. 

(c) Find $E\left[Y\right]$. 

Step-by-Step Solution

Answer

a. The $f(x,y)$ is a joint probability density function.

b. The value of $E\left(X\right)$ is $\frac{2}{5}$.  

c. The value of $E\left(Y\right)$ is $\frac{2}{5}$.

Step 1 : Given information

Let $f(x, y) = 24xy 0 \le x \le 1, 0 \le y \le 1, 0 \le x + y \le 1$ and let it equal 0 otherwise.

Step 2: Formula Used

The formula for a joint distribution said to be density function ,

${\int }_y{\int }_{x}f(x,y)dxdy=1$

Step 3 : Calculation (Part a)

Applying the formula to show is a joint probability density function.

$$\begin{gathered} {\int }_y{\int }_{x}f(x,y)dxdy={\int }_0^1{\int }_0^{1-x}24xydxdy \\ ={\int }_0^{1}24x{\left[\frac{y^2}{2}\right]}_0^{1-x}dx \\ ={\int }_0^{1}24x\left[\frac{1+x^2-2x}{2}\right]dx \\ ={\int }_0^{1}12\left(x+x^3-2x^2\right)dx \end{gathered}$$

Further integrating with respect to $dx$

$$\begin{gathered} ={12\left(\frac{x^2}{2}+{\frac{x}{4}}^4-\frac{2x^3}{3}\right)}_0^1 \\ =12\left(\frac{1}{2}+\frac{1}{4}-\frac{2}{3}\right) \\ =12\left(\frac{6+3-8}{12}\right) \\ =12\left(\frac{1}{12}\right) \\ =1 \end{gathered}$$

Conclusion: $f(x,y)$ is a joint probability density function$$\begin{gathered} {\int }_y{\int }_{x}f(x,y)dxdy=12{\left(\frac{x^2}{2}+\frac{x^4}{4}-\frac{2x^3}{3}\right)}^{1}0 \\ =12\left(\frac{1}{2}+\frac{1}{4}-\frac{1}{3}\right) \\ =12\left(\frac{2+3-4}{4\times 3}\right) \\ =1 \end{gathered}$$

Step 1: Find E ( X ) (Part b)

To find$E\left(X\right)$, first write the expression for $f_X\left(x\right)$,

$f_X\left(x\right) = {\int }_y{f}_{X,Y}(x,y)dxdy={\int }_0^{1-x}24xydy=12x{(1-x)}^2$when $x\ge 0$and zero otherwise.

$$\begin{gathered} \therefore E\left(X\right)={\int }_0^{1}x{f}_X\left(x\right)dx=12{\int }_0^1{x}^2{(1-x)}^{2}dx =12{\int }_0^1(x^2-2x^3+x^4)dx \\ =12\left(\frac{1}{3}-\frac{1}{2}+\frac{1}{5}\right)=\frac{2}{5} \end{gathered}$$

Step 1: Find E ( Y ) (part c)

By symmetry, $f_Y\left(y\right)=12y{(1-y)}^2$when $y\ge 0$ and zero otherwise

Since the Probability density functions are same, therefore by symmetry, the expectations will be equal.