4. The discrete random variables X and Y have known joint probability mass function с x = 0,1,2,L,9, y = 0,1,2,L,9, y≥x 0 where c = 1/55 otherwise a) Obtain the conditional PMF P(x/ Y = y) of the random variable X. b) Obtain the conditional PMFs P₂(y/X=x). c) Determine E{X/Y = 5} and var{X/Y = 5}. d) Compare the results of part c) with the same results when there is no conditioning. Py(x, y) given by Pxy(x, y) =< 3
4. The discrete random variables X and Y have known joint probability mass function с x = 0,1,2,L,9, y = 0,1,2,L,9, y≥x 0 where c = 1/55 otherwise a) Obtain the conditional PMF P(x/ Y = y) of the random variable X. b) Obtain the conditional PMFs P₂(y/X=x). c) Determine E{X/Y = 5} and var{X/Y = 5}. d) Compare the results of part c) with the same results when there is no conditioning. Py(x, y) given by Pxy(x, y) =< 3
A First Course in Probability (10th Edition)
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ISBN:9780134753119
Author:Sheldon Ross
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Chapter1: Combinatorial Analysis
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![4. The discrete random variables \(X\) and \(Y\) have a known joint probability mass function \(P_{XY}(x, y)\) given by
\[
P_{XY}(x, y) =
\begin{cases}
c & \text{if } x = 0, 1, 2, \ldots, 9, \, y = 0, 1, 2, \ldots, 9, \, y \geq x \\
0 & \text{otherwise}
\end{cases}
\]
where \(c = \frac{1}{55}\).
a) Obtain the conditional PMF \(P_X(x \mid Y = y)\) of the random variable \(X\).
b) Obtain the conditional PMFs \(P_Y(y \mid X = x)\).
c) Determine \(E\{X \mid Y = 5\}\) and \(\text{var}\{X \mid Y = 5\}\).
d) Compare the results of part c) with the same results when there is no conditioning.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9328926a-b343-4d47-8a00-cbd9e6e36ac2%2F2b2b8ee7-d396-46a8-964e-a28cec94137a%2F2ezt83o_processed.png&w=3840&q=75)
Transcribed Image Text:4. The discrete random variables \(X\) and \(Y\) have a known joint probability mass function \(P_{XY}(x, y)\) given by
\[
P_{XY}(x, y) =
\begin{cases}
c & \text{if } x = 0, 1, 2, \ldots, 9, \, y = 0, 1, 2, \ldots, 9, \, y \geq x \\
0 & \text{otherwise}
\end{cases}
\]
where \(c = \frac{1}{55}\).
a) Obtain the conditional PMF \(P_X(x \mid Y = y)\) of the random variable \(X\).
b) Obtain the conditional PMFs \(P_Y(y \mid X = x)\).
c) Determine \(E\{X \mid Y = 5\}\) and \(\text{var}\{X \mid Y = 5\}\).
d) Compare the results of part c) with the same results when there is no conditioning.
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VIEWStep 2: Obtain the conditional PMF P(x/ Y = y) of the random variable X
VIEWStep 3: Obtain the conditional PMFs P(y/ X = x)
VIEWStep 4: Determine E{X / Y = 5} and var{X / Y =5}
VIEWStep 5: Compare the results of part c) with the same results when there is no conditioning.
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