Prove the formulas given in the table at the beginning of Section 3.4 for the Bernoulli,...
8.3. Compute the measurement signal-to-noise ratio–that is, lul/o, where u = E[X] and o2 = Var(X)-of the following random variables: a. Poisson with mean 1; b. binomial with parameters n and p; c. geometric with mean 1/p; d. uniform over (a, b); e. exponential with mean 1/2; f. normal with parameters u, o?.
Please show steps (and formulas) for part b Problem 2. a. X has a normal distribution with mean 5 and variance 25. Y has a normal distribution with mean 3 and variance 16. In addition, X and Y are independent. If W = X+Y, find P(W > 9). b. Random variables U, V, Z are such that E[U] = 1, E[V] =5, E[2] = -3, Var[U] = 1, Var[V] = 4, Var[2] =1, Cov[U,V] =-1,Cov[U, 2] = 2, Cor[V, 2]...
6. (a) Given that X and Y are continuous random variables, prove from first principles that: (b) The random variable X has a gamma distribution with parameters-: 3 and A-2 . Y is a related variable with conditional mean and variance of =x)= Calculate the unconditional mean and standard deviation of Y. (c) Suppose that a random variable X has a standard normal distribution, and the conditional distribution of a Poisson random variable Y, given the value ol XOx, has...
3) Suppose X,,X,,X, (n > 1) is a random sample from Bernoulli distribution with Circle out your Class: Mon&Wed or Mon.Evening p.mf. p(x)=p"(I-p)'-x , x = 0,1, , thenyi follows ( ). Binomial distribution B(a.p) eNormal distribution N(p,mp(- O Poisson distribution P(np) Dcan not be determined. 4) Suppose X-N(0,1) and Y~N(24), they are independent, then )is incorrect. X+Y N(2, 5) C X-Y-NC-2,5) BP(Y <2)>0.5 D Var(X) < Var(Y) x,X,, ,X, (n>1) is a random sample from N(μσ2), let-1ΣΧί 5) Suppose...
In each of the summer months (June, July, August), the number of accidents per months at a busy intersection is Poisson distributed with mean 1.5 accidents/month. For all other months, the number of accidents is Poisson distributed with mean 0.5 accidents/month. 7. a) (3 pts) First, let Y an, YFeb, YMar,... be the number of accidents occurring in the months of January, February, March, etc. Define a variable A- the total number of accidents occurring in the second half of...
(2) Given two independent variables X1 and X2 having Bernoulli distribution with parameter p=1/3, let Y1 = 2X1 and Y2 = 2X2. Then A E[Y1 · Y2] = 2/9 BE[Y1 · Y2] = 4/9 C P[Y1 · Y2 = 0) = 1/9 D P[Y1 · Y2 = 0) = 2/9 (3) Let X and Y be two independent random variables having gaussian (normal) distribution with mean 0 and variance equal 2. Then: A P[X +Y > 2] > 0.5 B...
LU 22 2. We know that the sample variance follows a chi-square distribution: Sanx?(n-1). (a) (5 points) Use this fact to show that E(S) = 02. (Hint: Find the mean of the x as then mean of a Gamma distribution.) (b) (5 points) Use Markov's inequality to find an upper bound on the probability that the sample variance is twice the true variance, i.e. P(S? > 20%).
8. You are given two boxes, one contains nuts and the other contains bolts. Below is a picture of a bolt. The D indicates Below right is a side and overhead picture of a nut. The the diameter of the bolt D indicates the diameter of the hole INSIDE the nut. ATI RODI c ISO METRIC AND WASHERS A bolt is supposed to fit inside a nut. On the right is a picture of a bolt properly fitting inside a...
Problem 1 Let Xi, ,Xn be a random sample from a Normal distribution with mean μ and variance 1.e Answer the following questions for 8 points total (a) Derive the moment generating function of the distribution. (1 point). Hint: use the fact that PDF of a density always integrates to 1. (b) Show that the mean of the distribution is u (proof needed). (1 point) (c) Using random sample X1, ,Xn to derive the maximum likelihood estimator of μ (2...
Suppose X and Y are independent and Prove the following a) U=X+Y~gamma(α + β,γ) b) V=X/(X + Y ) ∼ beta(α,β) c) U, V independent d) ~gamma(1/2, 1/2) when W~N(0,1) X ~ gammala, y) and Y ~ gamma(6, 7) We were unable to transcribe this image