Consider a joint probability density function of two random variables X and Y…
2024
Consider a joint probability density function of two random variables X and Y
\(f_{X,Y}(x, y) = \begin{cases} 2xy, & \text{if } 0 < x < 2 \text{ and } 0 < y < x \\ 0, & \text{otherwise} \end{cases} \)
Then, 𝐸[𝑌|𝑋 = 1.5] is ______.
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Correct answer: 1
Step 1: Compute the marginal density of X: f_X(x) = ∫_0^x 2 x y dy = 2x · (x^2/2) = x^3 for 0 < x < 2.
Step 2: Find the conditional density of Y given X = x: f_{Y|X}(y|x) = f_{X,Y}(x,y) / f_X(x) = (2 x y) / x^3 = 2y / x^2, for 0 < y < x.
Step 3: Compute the conditional expectation E[Y | X = x]: E[Y | X = x] = ∫_0^x y · (2y / x^2) dy = (2 / x^2) ∫_0^x y^2 dy = (2 / x^2) · (x^3 / 3) = 2x / 3.
Final step: Evaluate at x = 1.5: E[Y | X = 1.5] = 2 · 1.5 / 3 = 1.
Answer: 1.