The joint probability function of two discrete random variables X and Y is g ven

Question

The joint probability function of two discrete random variables X and Y is g ven by f x; y c 2x + y where x and y can assume all inte ers such that S S S 3 and f(x; y)0 otherwise (a) Find the value of the constant c. Give your answer to three decimal places. 0.024 (b) Find P(x-o,Y-3). Give your answer to three decimal places. 0.524 (c) Find P(X2 1,Ys 0). Give your answer to three decimal places. (d) X and Y are independent random variables. Can't be determined False True Submit Answer Save Progress 3. + Question Details My Notes Define the joint pmf of (X, Y) by (o, 10)0, 20) 4/24, , 10) f(1, 30) 4/24, 1, 20)7 24, (2, 30)-1 24 Find the value of the folowing. Give your answer to three decimal places. c) E(Y|X= 2)= d) E(Y

Explanation / Answer

f(x,y) = c(2x + y)
(a)
Sum f(x,y) over all x and all y and the result should be 1.
c(2.0 + 0) + c(2.0 + 1) + c(2.0 + 2) + c(2.0 + 3) + c(2.1 + 0) + c(2.1 + 1) + c(2.1 + 2) + c(2.1 + 3) + c(2.2 + 0) + c(2.2 + 1) + c(2.2 + 2) + c(2.2 + 3) = 1
c + 2c + 3c + 2c + 3c + 4c + 5c + 4c + 5c + 6c + 7c = 1
42c = 1 => c = 0.024
(b)
P(X=0, Y=3) = 0.024 ( 2.0 + 3) = 0.072
(c)
P(X >= 1, Y <=0) = P(X >= 1, Y=0) = c(2.1+0) + c(2.2 + 0) = 6c = 6 x 0.024 = 0.144
d)
False
3.
f(0,10) = f(0,20) = 4/24
f(1,10) = f(1,30) = 4/24
f(1,20) = 7/24
f(2,30) = 1/24

a.
E[Y | X=0] = yf(x=0,y)
E[Y | X=0] = 10f(0,10) + 20f(0,20) = 10 x 4/24 + 20 x 4/24 = 5

b.
E[Y | X = 1] = yf(x=1,y)
E[Y | X = 1] = 10f(1,10) + 30f(1,30) + 20f(1,20) = 10 x 4/24 + 30 x 4/24 + 20 x 7/24 = 12.5
c.
E[Y | X = 2] = yf(x=2,y)
E[Y | X = 2] = 30f(2,30) = 30 x 1/24 = 1.25
d.
E[Y] = E[Y | X=0] + E[Y | X=1] + E[Y | X=2] = 5 + 12.5 + 1.25 = 18.75

4.
P(dot sent) = 3/7
P(dash sent) = 4/7
P(dot recieved | dash sent) = P(dash recieved | dot sent) = 1/10
P(dot recieved | dot sent) = P(dash recieved | dash sent) = 9/10
a)
P(dot recieved) = P(dot sent).P(dot recieved | dot sent) +
P(dash sent).P(dot recieved | dash sent)
P(dot recieved) = (3/7 x 9/10) + (4/7 x 1/10) = 0.4429
b)
P(dot sent | dot recieved) = P(dot sent dot recieved) / P(dot recieved)
P(dot recieved) = 0.4429
P(dot sent dot recieved) = P(dot sent) . P(dot recieved | dot sent)
P(dot sent dot recieved) = 3/7 x 9/10 = 0.3857
P(dot sent | dot recieved) = 0.3857 / 0.4429 = 0.8710

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