Examples: Binomial distribution ■ X ~ Exp(β), ■ f(x) ■ F(x) x > 0. E(X) ■ Var(X) ■ Moment generating function and first moment.
Suppose X∼Exp(λ) for some λ >0. Compute E(X) and Var(X).
2-t 2. Suppose that X - Exp(2), for some i >0. We know that the moment generating function of X is given by M(t)= E[e"]=-4, for some appropriate set of values of t. (a) Derive this mgf result. Explain what condition ont is necessary for this expression to be valid, and why this condition is necessary. (b) Use the mgf to find the first four moments ( u u , and ) of X. (c) Use your results in part...
Suppose that X - Exp(2), for some a > 0. We know that the moment generating function of X is given by M(O)= E[em]= , for some appropriate set of values of t. (a) Derive this mgf result. Explain what condition ont is necessary for this expression to be valid, and why this condition is necessary. (b) Use the mgf to find the first four moments (u, us, and u) of X. (c) Use your results in part (b) to...
2. Desk check this code. var sum = 0; for (var i=1; i <=3; i++) if ((i % 2) 0) { sum += i; else sum-= 1; Desk Check i % 2 sum
,X, ,n. independent, the central Xi, E(X)=0, var(X)-σ are Prove 3. Assume <o。 13<oo, 1=1, limit theorem (CLT) based EX1 result regarding what are conditions on σ that we need to assume in order for the x.B.= Σσ, as n →oo. In this context, X,, B" =y as n →oo, In this context, result to hold?
Prove, Var(ay)= a^2 var(y) Var(y+a)= var(y) Var(x+y)= var(x)+var(y)+2cov(x,y)
The following ψ are m.gf of some random variable X. Find EN] and Var(X) in each case. (a) ψ(t) = e2(e-1),-oo < t < oo.
2 (1) For z E C, define exp z - n-0 (a) Prove that the infinite series converges absolutely for z E C (b) Prove that when z є R, the definition of exp z given above is consistent with the one given in problem (2a), assignment 16. Definition from Problem (2a): L(x(1/t)dt E(z) = L-1 (z)
2 (1) For z E C, define exp z - n-0 (a) Prove that the infinite series converges absolutely for z E C...
3. (a) Given n e N, prove that sup{.22 : 0<x<1} = 1 and inf{.22n: 0<x<1} = 0. (b) Find the supremum of the set S = {Sn: ,ne N}. Give a proof.