5. Let X ~ N(m, σ) be a scalar Gaussian random variable with mean m and...
3. (10 pts.) X is a Gaussian random variable with E{X} = 2 and Var(X) = 16. Let Y = 3X +1. Determine the probability: Pr(Y > 2)
Problem 1. Let X be a contiuous random variable with probability density 2T f0SS Let A be the event that X > 1/2. Compute EXA) and Var(XA).
2. Assume X is a random variable following from N(μ, σ2), where σ > 0. (a) Write down the pdf of X (b) Compute E(X2). (b) Define YFind the distribution of Y.
Let > 0 and a > 0 be given. Suppose that X is a random variable with moment generating function e My(t) = {(A-ta tsy Top til Compute Var(X). Show that if we define Ly(t) = In My(t) then Ls (0) = Var(X).
Problem 1 For Gaussian distribution ρ (x)-ae Find: (1) Constant a; (2) <x> , <x> and standard deviation of the distribution; (3) Sketch the graph p(x) (x-b)2 -T2
Let X be an exponentially distributed random variable with parameter λ. Prove that P(X > s + tK > t) P(X > s) for any S,12 0
4. Let X be continuous random variable whose PDF is given by for r>1 otherwise, where 0 is an unknown scalar parmeter. (a) Find the maximum likelihood estimator of . (b) Find a method-of-moments estimator of θ for the case when θ > 1. (c) Why can we not find a method-of-moments estimator when θ < 1? 151 151 151
IF Let x(t) Show that e 20" σ>0, and let (o) be the Fourier transform of x(t) .
I. (5 points) Let X be a random variable with moment generating function M(t) = E [etx]. For t > 0 and a 〉 0, prove that and consequently, P(X > a inf etaM(t). t>0 These bounds are known as Chernoff's bounds. (Hint: Define Z etX and use Markov inequality.)
Q2. Assume that X is a continuous and nonnegative random variable with the cumulative distribution function Fx Let b> 0. (a) Find the cumulative distribution function of Y = XI(X < b} (b) Apply the general formula from (a) to exponential distribution with parameter λ > 0.