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. Warid 5:22 PM © 60% Х Assignment Number 3 Assignment Number 3 Q#1 Let X...

Question:

. Warid 5:22 PM © 60% Х Assignment Number 3 Assignment Number 3 Q#1 Let X and Y are joint probability functions given by xy2
. Warid 5:22 PM © 60% Х Assignment Number 3 Assignment Number 3 Q#1 Let X and Y are joint probability functions given by xy2 a- f(x, y) = ; x = 1,2,3; y = 1,2 b- f(x,y) = ; x = 2,4,5; y = 1,2,3 30 ху 66 Find the marginal probability functions of r.v X&Y also find out if X & Y are independent?

Answers

1.

(a)

Marginal probability distribution of X is as follows.

\tiny f\left ( x \right )=\begin{cases} \frac{1*1^2}{30}+\frac{1*2^2}{30} & \text{ if } x=1 \\ \\ \frac{2*1^2}{30}+\frac{2*2^2}{30} & \text{ if } x=2 \\ \\ \frac{3*1^2}{30}+\frac{3*2^2}{30} & \text{ if } x=3 \end{cases} =\begin{cases} \frac{5}{30} & \text{ if } x=1 \\ \\ \frac{10}{30} & \text{ if } x=2 \\ \\ \frac{15}{30} & \text{ if } x=3 \end{cases} =\begin{cases} \frac{1}{6} & \text{ if } x=1 \\ \\ \frac{2}{6} & \text{ if } x=2 \\ \\ \frac{3}{6} & \text{ if } x=3 \end{cases}

Marginal probability distribution of Y is as follows.

\tiny f\left ( y \right )=\begin{cases} \frac{1*1^2}{30}+\frac{2*1^2}{30}+\frac{3*1^2}{30} & \text{ if } y=1 \\ \\ \frac{1*2^2}{30}+\frac{2*2^2}{30}+\frac{3*2^2}{30} & \text{ if } y=2 \end{cases} =\begin{cases} \frac{6}{30} & \text{ if } y=1 \\ \\ \frac{24}{30} & \text{ if } y=2 \end{cases} =\begin{cases} \frac{1}{5} & \text{ if } y=1 \\ \\ \frac{4}{5} & \text{ if } y=2 \end{cases}

We observe that,

\tiny f\left ( x=1,y=1 \right )=\frac{1}{30}=\frac{1}{6}*\frac{1}{5}=f\left ( x=1 \right )*f\left ( y=1 \right )

\tiny f\left ( x=1,y=2 \right )=\frac{4}{30}=\frac{1}{6}*\frac{4}{5}=f\left ( x=1 \right )*f\left ( y=2 \right )

\tiny f\left ( x=2,y=1 \right )=\frac{2}{30}=\frac{2}{6}*\frac{1}{5}=f\left ( x=2 \right )*f\left ( y=1 \right )

\tiny f\left ( x=2,y=2 \right )=\frac{8}{30}=\frac{2}{6}*\frac{4}{5}=f\left ( x=2 \right )*f\left ( y=2 \right )

\tiny f\left ( x=3,y=1 \right )=\frac{3}{30}=\frac{3}{6}*\frac{1}{5}=f\left ( x=3 \right )*f\left ( y=1 \right )

\tiny f\left ( x=3,y=2 \right )=\frac{12}{30}=\frac{3}{6}*\frac{4}{5}=f\left ( x=3 \right )*f\left ( y=2 \right )

Hence, X and Y are independent.

(b)

Marginal probability distribution of X is as follows.

\tiny f\left ( x \right )=\begin{cases} \frac{2*1}{66}+\frac{2*2}{66}+\frac{2*3}{66} & \text{ if } x=2 \\ \\ \frac{4*1}{66}+\frac{4*2}{66}+\frac{4*3}{66} & \text{ if } x=4 \\ \\ \frac{5*1}{66}+\frac{5*2}{66}+\frac{5*3}{66} & \text{ if } x=5 \end{cases} =\begin{cases} \frac{12}{66} & \text{ if } x=2 \\ \\ \frac{24}{66} & \text{ if } x=4 \\ \\ \frac{30}{66} & \text{ if } x=5 \end{cases} =\begin{cases} \frac{2}{11} & \text{ if } x=2 \\ \\ \frac{4}{11} & \text{ if } x=4 \\ \\ \frac{5}{11} & \text{ if } x=5 \end{cases}

Marginal probability distribution of Y is as follows.

\tiny f\left ( y \right )=\begin{cases} \frac{2*1}{66}+\frac{4*1}{66}+\frac{5*1}{66} & \text{ if } y=1 \\ \\ \frac{2*2}{66}+\frac{4*2}{66}+\frac{5*2}{66} & \text{ if } y=2 \\ \\ \frac{2*3}{66}+\frac{4*3}{66}+\frac{5*3}{66} & \text{ if } y=3 \end{cases} =\begin{cases} \frac{11}{66} & \text{ if } y=1 \\ \\ \frac{22}{66} & \text{ if } y=2 \\ \\ \frac{33}{66} & \text{ if } y=3 \end{cases} =\begin{cases} \frac{1}{6} & \text{ if } y=1 \\ \\ \frac{1}{3} & \text{ if } y=2 \\ \\ \frac{1}{2} & \text{ if } y=3 \end{cases}

We observe that,

\tiny f\left ( x=2,y=1 \right )=\frac{2}{66}=\frac{2}{11}*\frac{1}{6}=f\left ( x=2 \right )*f\left ( y=1 \right )

\tiny f\left ( x=2,y=2 \right )=\frac{4}{66}=\frac{2}{11}*\frac{1}{3}=f\left ( x=2 \right )*f\left ( y=2 \right )

\tiny f\left ( x=2,y=3 \right )=\frac{6}{66}=\frac{2}{11}*\frac{1}{2}=f\left ( x=2 \right )*f\left ( y=3 \right )

\tiny f\left ( x=4,y=1 \right )=\frac{4}{66}=\frac{4}{11}*\frac{1}{6}=f\left ( x=4 \right )*f\left ( y=1 \right )

\tiny f\left ( x=4,y=2 \right )=\frac{8}{66}=\frac{4}{11}*\frac{1}{3}=f\left ( x=4 \right )*f\left ( y=2 \right )

\tiny f\left ( x=4,y=3 \right )=\frac{12}{66}=\frac{4}{11}*\frac{1}{2}=f\left ( x=4 \right )*f\left ( y=3 \right )

\tiny f\left ( x=5,y=1 \right )=\frac{5}{66}=\frac{5}{11}*\frac{1}{6}=f\left ( x=5 \right )*f\left ( y=1 \right )

\tiny f\left ( x=5,y=2 \right )=\frac{10}{66}=\frac{5}{11}*\frac{1}{3}=f\left ( x=5 \right )*f\left ( y=2 \right )

\tiny f\left ( x=5,y=3 \right )=\frac{15}{66}=\frac{5}{11}*\frac{1}{2}=f\left ( x=5 \right )*f\left ( y=3 \right )

Hence, X and Y are independent.

.

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