a) Find k b) Calculate P(X ≥ 1)
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- If a game gives payoffs of $10 and $100 with probabilities 0.9 and 0.1, respectively, then the expected value of this game is E=0.9+0.1= .Assume that the probability that an airplane engine will fail during a torture test is 12and that the aircraft in question has 4 engines. Construct a sample space for the torture test. Use S for survive and F for fail.If X is a continuous random variable with X ∼ Uniform([0, 2]), what is E[X^3]?
- X is a continuous random variable taking values between 0 and 2. If F(1.5) = 0.70 and P(X<1) = 0.20, what is P (1 < X < 1.5)?Suppose X is a continuous random variable with p.d.f. fX(x) = kx2(1 − x) if 0 < x < 1. (b) Find the c.d.f FX(x) explicitly.(b) Let Z be a discrete random variable with E(Z) = 0. Does it necessarily follow that E(Z³) = 0? If yes, give a proof; if no, give a counterexample.
- Suppose Xis a random variable with E(X) = 2, and E(X2) = 5. Compute the following quantities: a) Var(X) b) E(X−1)2 c) Var(−X−3) d) E(5X+ 4)If X1, X2, and X3 constitute a random sample of sizen = 3 from a Bernoulli population, show that Y =X1 + 2X2 + X3 is not a sufficient estimator of θ. (Hint:Consider special values of X1, X2, and X3.)
