Question

The joint probability distribution of two random variables X and Y is given below. Find the marginal distributions and P(Y=3/X=2)
f(x,y) x
y 2 4
1 0.1 0.15
3 0.2 0.3
5 0.1 0.15

Answer 1

Answer on Question #69192 – Math – Statistics and Probability

Question

The joint probability distribution of two random variables $X$ and $Y$ is given below. Find the marginal distributions and $P(Y=3/X=2)$

$$f(x,y) \times$$ $$\begin{array}{cccc} y & 2 & 4 \\ 1 & 0.1 & 0.15 \\ 3 & 0.2 & 0.3 \\ 5 & 0.1 & 0.15 \\ \end{array}$$

Solution

Marginal distribution of $X$:

$$f_x(2) = P(X = 2) = 0.1 + 0.2 + 0.1 = 0.4$$ $$f_x(4) = P(X = 4) = 0.15 + 0.3 + 0.15 = 0.6$$

Marginal distribution of $Y$:

$$f_y(1) = P(Y = 1) = 0.1 + 0.15 = 0.25,$$ $$f_y(3) = P(Y = 3) = 0.2 + 0.3 = 0.5,$$ $$f_y(5) = P(Y = 5) = 0.1 + 0.15 = 0.25$$$$P(Y=3/X=2) = \frac{P(X=2 \cap Y=3)}{P(X=2)} = \frac{f(2,3)}{f_x(2)} = \frac{0.2}{0.4} = \frac{1}{2} = 0.5.$$

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