X and Y are discrete random variables. The simultaneous probability distribution

Question

X and Y are discrete random variables. The simultaneous probability distribution, P (X = x Y = y) is given as follows:

a) There is a special interest in the variable Z = 2X - XY. Find the probable distribution of Z and E (Z) (hint: look at all possible values Z can take).

b) Use the simultaneous probability distribution to show that X and Y are not independent.

b) If two variables are independent then the covariance of the variables is equal to zero. If X and Y are independent, then the simultaneous probability distribution does not apply. What does E (Z) become if X and Y are independent?

x y y y y 0 1/8 1/4 1/8 0 1 0 1/8 1/4 1/8

Explanation / Answer

The given simultaneous probability distribution P(X=xY=y) of two random variables X and Y are as follows:

(a) The random variable Z is defined to be Z=2X-XY.

So, the possible values that the random variable Z can take are:

When X=0 and Y=y,

Z = 2*0-0*y = 0 and;

when X=1 and Y=y,

Z=2*1-1*y = (2-y)

So, the probability distribution on Z is given by:

ANd the expected value of Z is given by:

E(Z)=0x1/2 + (2-y)*1/2 = 1-y/2

Thus, E(Z) = 1-y/2

(b) For two events A and B to be independent it is required that P(AB)=P(A)*P(B).

Now, P(X=0)=1/2 and P(Y=y)=1. So, P(X=0)*P(Y=y)=1/s

Now, lets consider the first cell P(X=0Y=y)=1/8

Clearly, P(X=0)*P(Y=y)P(X=0Y=y)

Thus, we can say that the two random variables X and Y are not independent.

(c) When the two variables X and Y are independent the E(XY)=E(X)E(Y)

So, E(Z) = E(2X-XY) = 2E(X)-E(X)E(Y)

From the above tables it can be calculated that E(X)=1/2 and E(Y)=y

So. E(Z) = 2*1/2 -1/2*y = 1-y/2 which is the same as above.

x y y y y 0 1/8 1/4 1/8 0 =1/2 1 0 1/8 1/4 1/8 =1/2

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