Question

In: Statistics and Probability

Let X be a discrete random variable that takes value -2, -1, 0, 1, 2 each...

Let X be a discrete random variable that takes value -2, -1, 0, 1, 2 each with probability 1/5.

Let Y=X2

a) Find the possible values of Y. Construct a joint probability distribution table for X and Y. Include the marginal probabilities.

b) Find E(X) and E(Y).

c) Show that X and Y are not independent.

Solutions

Expert Solution

a) for Y is mapped on X

therefore possible value of X are {(-2)2,(-1)2,02,12,22} ={ 4,1,0,1,4) ={0,1,4}

below is joint probability distribution of X and Y:

y P(y)
x 0 1 4 Total
-2 0        0           1/5    1/5
-1 0           1/5 0           1/5
0    1/5 0        0           1/5
1 0           1/5 0           1/5
2 0        0           1/5    1/5
P(x)    1/5    2/5    2/5 1.0000

margnial probabilities are in P(X) row and P(y) column

b)

E(X)=-2*(1/5)+(-1)*(1/5)+0*(1/5)+1*(1/5)+2*(1/5)=0

E(Y)=0*(1/5)+1*(2/5)+4*(2/5)=2

c)

here as P(X=-2,Y=0)=0 while P(X=-2)*P(Y=0)=(1/5)*(!/5)=1/25

as they both are not equal; X and Y are not independent,


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