X is a random variable with a lognormal distribution and that Y = ln(X) ∼ N(µ, σ2 ). Prove that µX = e ^ (µ+ (σ^2)/2 )
X is a random variable with a lognormal distribution and that Y = ln(X) ∼ N(µ,...
5. A random variable X ∼ N (µ, σ2 ) is Gaussian distributed with mean µ and variance σ 2 . Given that for any a, b ∈ R, we have that Y = aX + b is also Gaussian, find a, b such that Y ∼ N (0, 1) Please show your work. Thanks!
. Suppose that Y is a normal random variable with mean µ = 3 and variance σ 2 = 1; i.e., Y dist = N(3, 1). Also suppose that X is a binomial random variable with n = 2 and p = 1/4; i.e., X dist = Bin(2, 1/4). Suppose X and Y are independent random variables. Find the expected value of Y X. Hint: Consider conditioning on the events {X = j} for j = 0, 1, 2. 8....
Let the random variable X follow a normal distribution with µ = 22 and σ2 = 7. Find the probability that X is greater than 10 and less than 17.
N (,02). We 7. A positive random variable Y is said to be a lognormal random variable, LOGN(1,02), if In Y assume that Y, LOGN(Mi, 0), i = 1,...,n are independent. [5] (a) Find the distribution of T = II Y. [4] (b) Find E(T) and Var(T) 5) (c) If we assume that Hi = ... = Hn and oi = ... = on what does the the successive geometric average, lim (IIYA), converge in probability to? Justify your answer....
3. Let Xi, , Xn be i.i.d. Lognormal(μ, σ2) (a) Suppose σ-1, prove that S-X(n)/X(i) is an ancillary statistics. (b) Suppose p 0, prove T-X(n) is a sufficient and complete statistics (c) Find a minimal sufficient statistics. 3. Let Xi, , Xn be i.i.d. Lognormal(μ, σ2) (a) Suppose σ-1, prove that S-X(n)/X(i) is an ancillary statistics. (b) Suppose p 0, prove T-X(n) is a sufficient and complete statistics (c) Find a minimal sufficient statistics.
Given a continuous random variable, prove that s--a:G-x) 2 converges to σ2 as Σ-1(xi-x) 2 converges to σ2 as n-1 Given a continuous random variable, prove that s--a:G-x) 2 converges to σ2 as Σ-1(xi-x) 2 converges to σ2 as n-1
7. A positive random variable Y is said to be a lognormal random variable, LOGN (u, 0), if In Y ~ N(No?). We assume that Y, LOGN (1,0%), i = 1,..., n are independent. [5] (a) Find the distribution of T = 11",Y. [4] (b) Find E(T) and Var(T) (5] (c) If we assume that M = ... = Hn and a = ... = 0, what does the the successive geometric average, lim (II",Y), converge in probability to? Justify...
A random variable X is said to follow a lognormal distribution if Y = log(X) follows a normal distribution. The lognormal is sometimes used as a model for heavy-tailed skewed distributions. please answer the follow: 110 15 60 5419 15 73 190 57 4344 18 37 43 55 19 23 82 175 50 80 65 63 36 6 10 17 52 43 70 22 95 20 4 17 15 12 29 29 6 22 40 17 26 30 16 116...
Let X1, ..., Xn be a random sample (i.i.d.) from a normal distribution with parameters µ, σ2 . (a) Find the maximum likelihood estimation of µ and σ 2 . (b) Compare your mle of µ and σ 2 with sample mean and sample variance. Are they the same?
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 Y.Find the distribution of Y