Question

Two descrebre rov X and Y have the following point p af x = 0, 1, 2 k (2x²y + 2y-x²-a f locry) = o otherwise y=113 Debemone 1 K (ii) bxy 1 Cor (XN) (iii) M (tutz) Use MCG, te) to confirm your answer to (11) Above Also assist here please?

kindly assist me to know how to find the E(x)=1.82...and E(Y)=1.16

Answer 1

The joint distribution is given by:
$f(x,y)=\begin{cases} \frac{1}{22}(x+y+1) & \text{ if } x=1,2;\;y=2 \\ \frac{1}{22}xy+\frac{1}{4} & \text{ if } x=2;\;y=0,1 \\0 & \text{otherwise} \end{cases}$

Now, the joint distribution is given by:

			Y
		0	1	2
	1	0	0	1 (x+y+1) 22
X	2	$\frac{1}{22}xy+\frac{1}{4}$	$\frac{1}{22}xy+\frac{1}{4}$	1 (x+y+1) 22

Putting in the values, we het the joint distribution as:

			Y
		0	1	2	Total
	1	0	0	0.181818	0.181818
X	2	0.25	0.340909	0.227273	0.818182
	Total	0.25	0.340909	0.409091	1

The Marginal Distribution of X is given by:

0.181818182 if x = 1 fx(x) = {0.818181818 if x = 2 0 otherwise

The Marginal Distribution of Y is given by:

$f_Y(y)=\begin{cases} 0.25 & \text{ if } y=0 \\ 0.340909091 & \text{ if } y=1 \\ 0.409090909 & \text{ if }y=2\\ 0 & \text{ otherwise } \end{cases}$

Now,

$=\sum_{x=1}^{2}x*P(X=x)$

$=1.818181818$
$=1.82$

Now,

$=\sum_{y=1}^{2}y*P(Y=y)$

$=1.159090909$
$=1.16$

Hence, the expectation of X and Y is proved to be 1.82 and 1.16 repectively.

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