Unformatted text preview: a. Verify that this is indeed a joint density function. b. Compute the density function of X . c. Find P [ X > Y ] d. Find P [ Y > ½| X < ½] e. Find E[ X ] and E[ Y ]. 3. If X and Y are independent exponential random variables with parameter λ = 1, compute the joint density of a. U = X + Y , V = X / Y ; b. U = X , V = X / Y ; c. U = X + Y , V = X / ( X + Y ). 4. A point is chosen at random (according to uniform a PDF) within the semicircle of the form Â Ã ±²³ ´µÄ·² » ¼·´ » Å Æ³ ´ Ç ¿È³· for some given r > 0. a. Find the joint PDF of the coordinates X and Y of the chosen point. b. Find the marginal PDF of Y and use it to find E[ Y ]. 5. The random variables X , Y , and Z are independent and uniformly distributed between zero and one. Find the PDF of X + Y + Z ....
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- Spring '11