Q1 Let X be a non-negative discrete random variable. Prove that the expected value of X satisfies E(X) = [ +∞o (1 - F(x)) dx
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Q: 1. Use Gauss-Jordan elimination to show that the matrix A = -1 -2 1 2 3 3 7 -12 is invertible,
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Q: What is the coefficient of x4y2z6 in the expansion of (3x² + 2y + 3z3)6?
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- 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.X1 and X2 are independent random variables such that Xi has PDF fXi(x)={λiexp(−λix) when x≥0, 0 otherwise}. What is P[X2<X1]?If X is a continuous random variable with X ∼ Uniform([0, 2]), what is E[X^3]?
- Let Y be a discrete random variable. Let c be a constant. PROVE Var (Y) = E (Y2) - E (Y)2Consider a function F (x ) = 0, if x < 0 F (x ) = 1 − e^(−x) , if x ≥ 0 Is the corresponding random variable continuous?Use the moment generating function technique to solve. Let X1, . . . , Xn be independent random variables, such that Xi ∼ Exponential(θ), for i =1, . . . , n. Find the distribution of Y = X1 + · · · + Xn.
- Let X be a random variable with pdf given by fX(x) = 1/[π(1 + x2)] for all real number x. Prove that X and 1/X are identically distributed by showing that they have the same probability distribution.Suppose the random variable y is a function of several independent random variables, say x1,x2,...,xn. On first order approximation, which of the following is TRUE in general?Let Y be a continuous random variable. Let c be a constant. PROVE Var (Y) = E (Y2) - E (Y)2
- Suppose that the random variable X is continuous and takes its values uniformly over the interval from 0 to 2. What is P{X = 1.5 or X = 0.4}?Let X1,...,Xn be iid random variables with expected value 0, variance 1, and covariance Cov [Xi,Xj] = ρ, for i≠j. Use Theorem of linearity of expectation to find the expected value and variance of the sum Y = X1 +...+Xn.For any continuous random variables X, Y , Z and any constants a, b, show the following from the definition of the covariance: