The moment-generating function of a random variable X is given by M(t) = 0.2 +0.4e5t +0.1e20t +0.3e¹⁰t (a) Find the pmf of X. (b) Find the values of E(X+3) and o² for X.

Could you solve (d),(e),(f)? Thank you.

Q1 Let X1 and X2 be independent exponential random variables with identical parameter A. Q1(i.) Find the distribution of Z = max(X1, X2). Q1 (ii.) Find the distribution of Y = min(X1, X2). Q1(iii.) Calculate E[Y]. Q1(iv.) Calculate E[Z]. Q1(v.) Using the relation Z = X1+X2 – Y, Calculate E[Z] and verify that it agrees with the calculation done in part (iv.)

Clearly hand write， As soon as possible

1. Let X have the moment-generating function Mx (t) = (1– 2t )r/2 , t< ½ (a) Find Var ( -5X) + E(X+2X²) (b) Find the moment-generating function of the random variable Z =X+2

The random variable U has the following distribution: U 123 PU(u) 0.20 0.40 0.40 (c) Write down the moment generating function (mgf) of U. (d) Use the mgf to help determine var[U].

A pseudo-random signal (S) of -3 or 3 is transmitted over a noisy channel. S=-3 is transmitted with probability of 1/3.  The received signal, Y, is the sum of the sent signal and the noise, N. The noise is modeled as a normal Gaussian with E[N]=0,var(N)=4 . The received signal is decoded as -3 for Y≤−0.5  and 3 for Y>−0.5. a. What is the probability of error in this system? b. To improve the system, each signal is transmitted three times and output is determine by majority rule (2 or more). What is the new probability of error?

The moment-generating function of the discrete random variable X is 1 M(t) =e 1 1 +. -2t 2t -3t e+ 6. . 3 What is the value of P(X> 1) ) 6.

Let X; be random variables and M; (t) be their respective moment generating functions, for i = 1, 2, 3. Solve: Suppose that M₁ (t) = 1 +t+t² + t³+... Is this enough information to determine the type of random variable that X₁ is? Explain why or why not. Determine the type of random variable if there is enough information. Determine if it is possible for M₂ (t) = (-1)" -0 (2n)! possible, determine the type of random variable that X₂. -+2n. Explain why or why not. If it is Suppose that that the expectation of X3 is -1 and its second moment is 1. Determine if this is enough information to find the moment generating function of X3 explicitly. Explain why or why not. If it is enough information, compute it explicitly.

Let X be an Exponential random variable, with the moment generating function (MGF): M(t)= (1-9t)-¹. (a) Derive the MGF above. (b) Derive E(X) using the MGF. (c) Derive V(X) using the MGF. (d) Find the MGF of Y, where Y = 2X-3.