a) Find the marginal density of "X" and the marginal density of "Y." (add them to the above table)
"P(X=1)={1 \\over 12}+{1 \\over 6}+{1 \\over 12}={1 \\over 3}"
"P(X=2)={1 \\over 12}+{1 \\over 12}+0={1 \\over 6}"
"P(X=3)={1 \\over 12}+0+{1 \\over 6}={1 \\over 4}"
"P(Y=2)={1 \\over 12}+{1 \\over 12}+{1 \\over 12}+{1 \\over 12}={1 \\over 3}"
"P(Y=3)={1 \\over 12}+{1 \\over 6}+{1 \\over12}+0={1 \\over 3}"
"P(Y=4)={1 \\over 12}+{1 \\over 12}+0+{1 \\over6}={1 \\over 3}"
b) Are "X" and "Y" independent?
To check whether "X" and "Y" are independent, we need to check that "P(X=x_i, Y=y_j)=P(X=x_i)P(Y=y_j)"
"i=1,2,3,4; j=1,2,3"
Thus, we conclude that "X" and "Y" are not independent.
c) Compute the P{Y>1| X>2}
"P(Y>1|X>2)={{1 \\over 12}+0+{1 \\over 6}\\over {1 \\over 4}}=1"
d) Compute the expected value of X.
e) Compute the probability that "X" is greater or equal to "Y+1."
f) Compute the conditional probability distribution of "X" given "Y=3."
"P(X=0|Y=3)={P(X=0,Y=3)\\over P(Y=3)}={1\/12\\over 1\/3}={1\\over4}"
"P(X=1|Y=3)={P(X=1,Y=3)\\over P(Y=3)}={1\/6\\over 1\/3}={1\\over2}"
"P(X=2|Y=3)={P(X=2,Y=3)\\over P(Y=3)}={1\/12\\over 1\/3}={1\\over4}"
"P(X=3|Y=3)={P(X=3,Y=3)\\over P(Y=3)}={0\\over 1\/3}=0"
"\\def\\arraystretch{1.5}\n \\begin{array}{c:c}\n x & 0 & 1 & 2 & 3 \\\\ \\hline\n f_{X|Y=3}(x) & 1\/4 & 1\/2 & 1\/4 & 0\n\\end{array}"
Comments
Leave a comment