1. Let X be a random variable. Is it true that X + X = 2X? Is its true that X ? X = 0? Why?
2x 0<x<1 Let X be a continuous random variable with probability density function f(x)= To else The cumulative distribution function is F(x). Find EX.
Let X be a continuous random variable with probability density function fX(x)=2x for 0 < x <1. What is the expected value of X.
X is a random variable uniformly distributed on [-3,1]. 1. Let Y = 2X – 1, find the pdf of Y. 2. Let Z = [X], find the pdf of Z. 3. What is the pdf of Y = [X + 3/?
Let X be a continuous random variable whose probability density function is fax) (2x +af (0 1) if x E (0; 1) f (x) - ind 1) the coefficient a; 2) P(0.5<X<0.7); 3) P(X 3) Part 3 sample of measurements is given XI-8 -210|2|8 Y 8 4 2 2 0
7. Let X be a continuous random variable with probability density function: 0, 5+2x if if 0 if x> 10 x<b f(x)=re, 76-0 5310 x 10 150 o Find the expected value and mode of random variable X. Part 3. Statistics The following sample is given 3.15 3.25 3.25 3.35 3.45 3.5 3.5
8. Let X be a continuous random variable with mgf given by It< 1 M(t)E(eX) 1 - t2 (a) Determine the expected value of X and the variance of X [3] (b) Let X1, X2, ... be a sequence of iid random variables with the same distribution as X. Let Y X and consider what happens to Y, as n tends to oo. (i) Is it true that Y, converges in probability to 0? (Explain.) [2] (ii) Explain why Vn...
Let X be exponential random variable with λ = 1. (a) Define Y = √ X. Specify the support of Y and find its density. (b)Define Z = X^2 + 2X. Specify the support of Z and find its density.
Let X be a random variable that is equal to 0 with probability 0.4 and to 1 with probability 0.6. Then, Select one: © A. Var(2X + 1) = 1.48 © B. Var(2X + 1) = 0.96 © C. Var(2X + 1) = 0.48 D. Var(2X + 1) = 1.96
Consider an exponentially distributed random variable X with pdf f(x) = 2e−2x for x ≥ 0. Let Y = √X. a. Find the cdf for Y. b. Find the pdf for Y. c. Find E[Y]. If you want to skip a difficult integration by parts, make a substitution and look for a Gamma pdf. d. This Y is actually a commonly used continuous distribution. Can you name it and identify its parameters? e. Suppose that X is exponentially distributed with...
7. Section 6.4, Exercise 1 Let X. X be a random sample from the U(0,0) distribution, and let , 2X and mx X, be estimators for 0. It is given that the mean and variance of oz are (a) Give an expression for the bias of cach of the two estimators. Are they unbiased? (b) Give an expression for the MSE of cach of the two estimators. (c) Com pute the MSE of each of the two ctrnators for n...