Concept explainers
Let
Want to see the full answer?
Check out a sample textbook solutionChapter 3 Solutions
EBK AN INTRODUCTION TO MATHEMATICAL STA
- If we let RX(t) = ln MX(t), show that R X(0) = μ and RX(0) = σ2. Also, use these results to find the mean and the variance of a random variable X having the moment-generating function MX(t) = e4(et−1)arrow_forwardX is an exponential random variable with λ =1 and Y is a uniform random variable defined on (0, 2). If X and Y are independent, find the PDF of Z = X-Y2arrow_forwardLet X be a Gaussian random variable (0,1). Let M = ln(5*X) be a derived random variable. What is E[M]?arrow_forward
- X is a discrete random variable and takes the values 0,1 and 2 with probabilities of 1/6, 1/3 and 1/2, respectively. What is the moment generator function M(t) of X?arrow_forward9.19 Let X and Y be two continuous random variables, with joint proba- bility density function f(x, y): - 30 -50x²-50y² +80xy for -arrow_forward(b) Let Z be a discrete random variable with E(Z) = 0. Does it necessarily follow that E(Z³) = 0? If yes, give a proof; if no, give a counterexample.arrow_forward
- Answer the following questions. Let X be a continuous random variable with P(X<0)=0. When E(X)=\mu exists, P(X\ge 3\mu) \le \frac{1}{(a)} by the Markov's inequality. What is (a)? Consider two random variables X and Z. The relationship between X and Z is given as X=1+2Z. Let Z be a random variable with moment generating function (mgf), M_Z(t) = (1-t)^{-3}, for t<1. What is the expectation of X?arrow_forwardLet X1...., Xn be a random sample of size n from an infinite population and assume X1 d= a + bU2 with the constants a > 0 and b > 0 unknown and U a standard uniform distributed random variable given by FU (x) := P(U ≤ x) = 0 if x ≤ 0 x if 0 < x < 1 1 if x ≥ 1 1. Compute the cdf of the random variable X1. 2. Compute E(X1) and V ar(X1). 3. Give the method of moments estimators of the unknown parameters a and b. Explain how you construct these estimators!arrow_forwardThis is a poisson mass function for the future lifetime of a newborn: f0(k) = λk e- λ/k! for all k>= 0,where k is a discrete random variable.For λ= 3 estimate F4(16)arrow_forward
- Let U1, ....U5 be independent and standard uniform distibuted random variables given by P(U1 ≤ x) = x, 0 < x < 1 1. Compute the moment generating function E(e sU ) of the random variable U1. 2. Compute the moment generating function of the random variable Y = aU1 + U2 + U3 + U4 + U5 with a > 0 unknown. 3. Compute E(Y ) and V ar(Y ). 4. As an estimator for the unknow value θ = a we migth use as an estimator θb = 2 n Xn i=1 Yi − 4 = 2Y − 4. with Yi independent and identically distributed having the same cdf as the random variable Y discussed in part 2. Compute E(θb) and V ar(θb) and explain why this estimator is sometimes not very useful. 5.Give an upperbound on the probability P(| θb− a |> ) for every > 0.(Hint:Use Chebyshevs inequality!)arrow_forwardLet X be a random variable with pdff(x) = 4x^3 if 0 < x < 1 and zero otherwise. Use thecumulative (CDF) technique to determine the pdf of each of the following random variables: 1) Y=X^4, 2) W=e^(-x) 3) Z=1-e^(-x) 4) U=X(1-X)arrow_forwardConsider a random variable, Y , which has a quasi-Bernoulli structure. With probability p ∈ [0, 1] it takes value 0. With probability (1 − p) it is described by a continuous random variable, X , with the following PDF, f_X (x)=1+x, x∈[−1,0), −1≤x<0f_X (x)=1−x, x∈[0,1], 0≤x≤1f_X (x)=0, otherwise , x>1 Obtain CDF of Y, F_Y (y), and draw the sketch.arrow_forward
- MATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons IncProbability and Statistics for Engineering and th...StatisticsISBN:9781305251809Author:Jay L. DevorePublisher:Cengage LearningStatistics for The Behavioral Sciences (MindTap C...StatisticsISBN:9781305504912Author:Frederick J Gravetter, Larry B. WallnauPublisher:Cengage Learning
- Elementary Statistics: Picturing the World (7th E...StatisticsISBN:9780134683416Author:Ron Larson, Betsy FarberPublisher:PEARSONThe Basic Practice of StatisticsStatisticsISBN:9781319042578Author:David S. Moore, William I. Notz, Michael A. FlignerPublisher:W. H. FreemanIntroduction to the Practice of StatisticsStatisticsISBN:9781319013387Author:David S. Moore, George P. McCabe, Bruce A. CraigPublisher:W. H. Freeman