given below is a bivariate distribution for the random variables x and y.\n f(x,y) x y\n 0.3 60 90\n 0.6 30…

Explanation:

Step1: Recall expected - value formula for discrete random variables

The expected value of a discrete random variable $X$ is $E(X)=\sum_{i}x_ip_i$. For $x$: $E(x)=0.3\times60 + 0.6\times30+0.1\times50=18 + 18+5=41$ For $y$: $E(y)=0.3\times90 + 0.6\times30+0.1\times60=27 + 18+6=51$

Step2: Recall variance formula for discrete random variables

The variance of a discrete random variable $X$ is $Var(X)=\sum_{i}(x_i - E(X))^{2}p_i$. For $x$: $(60 - 41)^{2}\times0.3+(30 - 41)^{2}\times0.6+(50 - 41)^{2}\times0.1$ $=19^{2}\times0.3+(- 11)^{2}\times0.6 + 9^{2}\times0.1$ $=361\times0.3+121\times0.6 + 81\times0.1$ $=108.3+72.6+8.1 = 189$ For $y$: $(90 - 51)^{2}\times0.3+(30 - 51)^{2}\times0.6+(60 - 51)^{2}\times0.1$ $=39^{2}\times0.3+(-21)^{2}\times0.6 + 9^{2}\times0.1$ $=1521\times0.3 + 441\times0.6+81\times0.1$ $=456.3+264.6 + 8.1=729$

Step3: Find probability distribution of $x + y$

When $x = 60,y = 90,x + y=150$ with probability $0.3$ When $x = 30,y = 30,x + y=60$ with probability $0.6$ When $x = 50,y = 60,x + y=110$ with probability $0.1$ So the probability distribution of $x + y$ is:

$x + y$	$f(x + y)$
$150$	$0.30$
$60$	$0.60$
$110$	$0.10$

Step4: Compute $E(x + y)$

$E(x + y)=150\times0.3+60\times0.6 + 110\times0.1=45+36 + 11=92$

Step5: Compute $Var(x + y)$

$(150 - 92)^{2}\times0.3+(60 - 92)^{2}\times0.6+(110 - 92)^{2}\times0.1$ $=58^{2}\times0.3+(-32)^{2}\times0.6+18^{2}\times0.1$ $=3364\times0.3 + 1024\times0.6+324\times0.1$ $=1009.2+614.4+32.4 = 1656$

Answer:

$E(x)=41$ $E(y)=51$ $Var(x)=189$ $Var(y)=729$

$x + y$	$f(x + y)$
$150$	$0.30$
$60$	$0.60$
$110$	$0.10$
$E(x + y)=92$
$Var(x + y)=1656$