Problem 2 (18 pts) =ez(z-1) Function pX Y(x y)=c(x+y) is a joint probability mass function over the nine points with x=123 and y=123 Then determine (a) The value of c (b) P(X=1 Y4) (c) P(X=1) (d) P(Y=2 X=1) (e) P(X3 Y=2) (f) The mean E(X) and variance Var(X) of the random variable X

Question

StudyX AI Fast Model · Accepted Answer

Solution By Steps

Step 1: Determine the value of c
Since it's a joint probability mass function, the sum of all probabilities over the nine points must equal 1.
Sum of probabilities = $c(1+1+2+2+3+3) = 12c$
Set this equal to 1: $12c = 1$
$c = \frac{1}{12}$

Step 2: Calculate $P(X=1, Y<4)$
$P(X=1, Y<4) = P(X=1, Y=1) + P(X=1, Y=2) + P(X=1, Y=3)$
$P(X=1, Y<4) = c(1+1) + c(1+2) + c(1+3)$
$P(X=1, Y<4) = \frac{1}{12}(2+3+4) = \frac{9}{12} = \frac{3}{4}$

Step 3: Calculate $P(X=1)$
$P(X=1) = P(X=1, Y=1) + P(X=1, Y=2) + P(X=1, Y=3)$
$P(X=1) = \frac{1}{12}(2+3+4) = \frac{9}{12} = \frac{3}{4}$

Step 4: Calculate $P(Y=2 \mid X=1)$
$P(Y=2 \mid X=1) = \frac{P(X=1, Y=2)}{P(X=1)}$
$P(Y=2 \mid X=1) = \frac{\frac{1}{12}(1+2)}{\frac{1}{12}(2+3+4)} = \frac{\frac{3}{12}}{\frac{9}{12}} = \frac{1}{3}$

Step 5: Calculate $P(X<3 \mid Y=2)$
$P(X<3 \mid Y=2) = P(X=1, Y=2) + P(X=2, Y=2)$
$P(X<3 \mid Y=2) = \frac{1}{12}(1+2) + \frac{1}{12}(2+2) = \frac{5}{12}$

Step 6: Calculate the mean $\mathrm{E}(X)$ and variance $\operatorname{Var}(X)$
$\mathrm{E}(X) = \sum_{x} x \cdot P(X=x)$
$\mathrm{E}(X) = 1 \cdot \frac{3}{4} + 2 \cdot \frac{3}{4} + 3 \cdot \frac{3}{4} = 2$

$\operatorname{Var}(X) = \mathrm{E}(X^2) - (\mathrm{E}(X))^2$
$\mathrm{E}(X^2) = 1^2 \cdot \frac{3}{4} + 2^2 \cdot \frac{3}{4} + 3^2 \cdot \frac{3}{4} = \frac{21}{4}$
$\operatorname{Var}(X) = \frac{21}{4} - 2^2 = \frac{5}{4}$

Final Answer
(a) $c = \frac{1}{12}$  
(b) $P(X=1, Y<4) = \frac{3}{4}$  
(c) $P(X=1) = \frac{3}{4}$  
(d) $P(Y=2 \mid X=1) = \frac{1}{3}$  
(e) $P(X<3 \mid Y=2) = \frac{5}{12}$  
(f) $\mathrm{E}(X) = 2$, $\operatorname{Var}(X) = \frac{5}{4}$

Key Concept
Joint Probability Mass Function

Key Concept Explanation
Joint probability mass functions describe the probabilities of multiple random variables taking specific values simultaneously. They are essential in understanding the combined behavior of random variables and are used in various fields such as statistics, machine learning, and decision theory.

Problem 2 18 pts ez z 1 Function pX Y x y c x y is a joint probability mass function over the nine points with x 123 and y 123 Then determine a The value of c b P X 1 Y4 c P X 1 d P Y 2 X 1 e P X3 Y 2...

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Problem 2 18 pts ez z 1 Function pX Y x y c x y is a joint probability mass function over the nine points with x 123 and y 123 Then determine a The value of c b P X 1 Y4 c P X 1 d P Y 2 X 1 e P X3 Y 2...

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