Question

3. (10pts) Let Y be a continuous random variable having a gamma probability distribution with expected value 3/2 and variance 3/4. If you run an experiment that generates one-hundred values of Y , how many of these values would you expect to find in the interval [1, 5/2]?

4. (10pts) Let Y be a continuous random variable with density function f(y) = 1 2 e −|y| , −∞ < y < ∞ 0, elsewhere (a) Find the moment-generating function of Y . (b) Use the moment-generating function you find in (a) to find the V (Y ).

5. (10pts) Let Y be a uniformly distribution random variable. Find: (a) P(|Y − µ| ≤ 2σ) (b) Use Tchebysheff’s theorem to estimate P(|Y − µ| ≤ 2σ). (c) Use the empirical rule to estimate P(|Y − µ| ≤ 2σ). (d) How does (b) compare to (a)? (e) How does (c) compare to (a)

Answer 1

For number 4, the pdf has constant 1/2 or 12 before it... since this is unsure hence please post it for part 4 and 5.