Let continuous random variables (W,X) have joint probability density function: S
ID: 2916092 • Letter: L
Question
Let continuous random variables (W,X) have joint probability density function: Sketch the support plane for the given joint distribution. Find the marginal distribution of W. Find P(X 3 OR W 3). Find P(W 2, X 1). Find the conditional distribution of X given W - w0, where 0Explanation / Answer
A) First quadrant in the xw plane (x > 0, w > 0) B) int(w * e^(-w*(x+1)), x = 0..) -e^(-w*(x+1)) evaluated at x = 0.. (-e^(-w*(+1))) - (-e^(-w*(0+1))) (-e^(-) + e^-w) e^-w C) P(X > 3 OR Y > 3) = P(X > 3) + P(Y > 3) - P(X > 3 AND Y > 3) int(int(w * e^(-w*(x+1)), x = 3..), y = 0..) +int(int(w * e^(-w*(x+1)), x = 0..), y = 3..) -int(int(w * e^(-w*(x+1)), x = 3..), y = 3..) Lots of integration which I'll leave to you.... D) int(int(w * e^(-w*(x+1)), x = 0..1), y = 0..2) A bit of calculus -e^(-2) + e^(-4)/2 + 1/2 E) Just plug in w0 for w. f(x) = w0 * e^(-w0 * (x+1)) F) int(1/2 * e^(-(x+1)/2), x = 0..2) -e^(-(x+1)/2) evaluated at x = 0..2 1 - e^(-3/2)
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