Suppose that X and Y are discrete random variables on the probability space (Q.F.P). Show that E[aX] = aE[X] and E[X+Y] = E[X] + E[Y].
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- Consider a random variable X taking the values k1, k2, , km E R .. with probability ; Pn E [0, 1] 1. Write down the formula for the P1, P2, .. respectively, where p1 + P2 + expected value of f(X) for a given function f(-). + Pn ...Suppose that W is a random variable with E(W4)< ∞. Show thatE(W2) < ∞.A bag of peanuts in their shells contains 149 peanuts. 57 of the shells contain one peanut, 88 of the shells contain two peanuts, and the rest contain three peanuts. Of the shells with one peanut, 14 of them are cracked. Of the shells with two peanuts, 24 of them are cracked. None of the shells with three peanuts are cracked. One peanut shell is randomly selected from the bag. What is the probability the shell is cracked? What is the probability the shell is not cracked and it contains two peanuts? What is the probability the shell is not cracked or it contains two peanuts? What is the probability the shell is not cracked given that it contains two peanuts?
- Let E and F be events in an experiment. If P(E|F)=0.5, P(E|F')=0.2 ,and P(E∩F')=0.1, find P(F) and P(E).If X and Y are Gaussian random variables then what is E[XY]?If A, u are the rates of arrival and departure in an M/M/1 queue respectively, give the formula for the probability that there are n customers in the queue at any time in the steady-state.
- The diameter of a round rock in a bucket, can be messured in mm and considered a random variable X in f(x) f(x) = k(x-x4) if 0 ≤ x ≤ 1f(x) = 0 otherwise. Find the expectation E(4X+ 3) and variance V(4X+3)9. If X and Y are two random variables and let g(X) be a random variable. Show that (a) E[g(X) X=x] = g(x). (b) E[g(x)Y|X=x] = g(x) E[Y|X=x]. Assume that E[g(x)] and E[Y] exist.Is P(S = s) = fs(s) = 1+11+1 ; s = 0,1,2,... 0; е. w. a discrete probability function? Why or why not?
- 3. Suppose it is known from large amounts of historical data that X, the number of cars that arrive at a specific intersection during a 20-second time period, is characterized by the following discrete probability function: 6x f(x) = e-6, for x = 0,1,2, ... a) Find the probability that in a specific 20-second time period, more than 8 cars arrive at the intersection.Theorem 11. Let X be a random variable and let g(x) be a non-negative function. Then for r > 0, Eg (X) P[g(X) > r] < Proof.Is the following valid? true or false 3r m(t) =0.6e + 0.le" +0.2e" %3D