Expert Answer to "The joint distribution of X and Y is..."

miyoko23q3

Answered question

2021-11-25

The joint distribution of X and Y is as follows.

\begin{array}{cc} V a l u e s o f Y \\ 1 & 0 & P {X = x} \\ V a l u e s o f X & 1 & 0.1 & 0.2 & 0.3 \\ 0 & 0.3 & 0.4 & 0.7 \\ P {Y = y} & 0.4 & 0.6 & 1.0 \end{array}

a. Find the marginal distribution of X and Y.
b. Find the conditional distribution of Y given

X - 1, P {Y = y ∣ X = 1}

c. Compute the conditional expectation of Y given

X = 1, E {Y = y ∣ X = 1}

Answer & Explanation

Charles Wee

Beginner2021-11-26Added 14 answers

a) Marginal distribution of X

\begin{array}{cc} x & 1 & 0 \\ P (X = x) & 0.3 & 0.7 \end{array}

Marginal distribution of Y

\begin{array}{cc} x & 1 & 0 \\ P (Y = y) & 0.4 & 0.6 \end{array}

P (Y ∣ X = 1)

P (Y = 0 ∣ X = 1) = \frac{P (X = 1, Y = 0)}{P (X = 1)}

= \frac{0.2}{0.3}

= 0.6667

P (Y = 1 ∣ X = 1) = \frac{P (X = 1, Y = 1)}{P (X = 1)}

= \frac{0.1}{0.3}

= 0.3333

E [Y ∣ X = 1]

= \sum_{y} y \times P (Y = y ∣ X = 1)

= 0 \times \frac{0.2}{0.3} + 1 \times \frac{0.1}{0.3}

= 0.3333

The required expectation is 0.3333.

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