In: Statistics and Probability
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.
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,