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- Suppose X is a discrete random variable and P(X = x) = (x+1)2 / C, for some positive constant C and for all x ∈ {0,1,3,4}. Solve for C and find the cdf of X.Let (X₂) be independent Bernoulli (1/2) random variables. Let Mk be the number of n E N with n ≤ k so that Xn Xn+1 = 1. Show that Mk k P 1 k→∞ 4 =Suppose X is a random variable taking possible values in {1,2,3,...} and 0 < P(X= 1) < 1, and that X satisfies the memoryless property. Prove that X must be a geometrically distributed random variable for some parameter value p.
- 1. 1. Let X;, i = 1, 2, ..., n be i.i.d. exponential random variables with mean 1. By the Central Limit Theorem, %3D Vn( X;/n – 1) → N(0, 1). i=1 Verify that the Lindeberg condition and the Lyapunov condition are satisfied.Suppose that X, Y, and Z are random variables with X ~ Bin(16, 1/4), Y ~ Geom(1/2), and Z ~ Poisson(5). Suppose further that X and Z are independent but that X and Y are not independent. Match up the following quantities. E(4Y – 27) Drag answer here 23 E(Y² + Z²) Cannot be determined Drag answer here Var(2X + Y) 36 Drag answer here -2 Var(X – Z) Drag answer here Var(X + 2Z) Drag answer here7
- Assume that X and Y are independent continuous random variables. Suppose that X in uniformly distributed on the interval [1,12] and Y is uniformly distributed on the interval [1,8]. Find O 0.01394 O 0.01533 O 0.01673 O 0.01255 O 0.01115Let X: 2 R be a discrete random variable on a discrete sample space 2 with distribution function Fx Select which of the following statements are true: The random variable -X has distribution function -Fx. Fx : R → R is a continuous function. O Fx is monotonically increasing. O limt00 Fx (t) = 0Q2 Suppose X1, X2, X3 are independent Bernoulli random variables with p = 0.5. Let Y; be the partial sums, i.e., Y1 = X1, Y2 = X1 + X2, Y3 = X1 + X2 + X3.