probability density function for a continuous random variable

Carnegie Mellon University Department of Mathematical Sciences 21256, Spring 2021 Exam 3, Part 2, Lecture 1 PART 2 1. Let 2 f (x) = kxe x ; if x 0 0; if x < 0 (a) Determine k such that f (x) is a probability density function for a continuous random variable X: (b) Determine the mean of X: 2. Suppose f (x; y) = k (x) (1 x) y (1 0; y) ; if (x; y) 2 [0; 1] [0; 1] otherwise (a) Determine k such that f (x) is a probability density function for continuous random variables X and Y: (b) Find the probability that X + Y 1: 3. Consider a waiting time problem for independent events T and U: Suppose the mean T for event T is 4 seconds, and the mean U for event U is 5 seconds. Calculate the probability that 8 T + U 10:
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