40 Let f(x) be any pdf with mean and variance o2. Show how to create a location-scalefamily based on f(x) such that the standard pdf of the family, say f*(x), has mean 0and variance 1.41 A family of cdfs {F(x0), 0 } is stochastically increasing in 0 if 0₁ > 02 ⇒ F(x|0₁)is stochastically greater than F(x|02). (See Exercise 1.49 for the definition of stochas-tically greater.)

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Answered: 40 Let f(x) be any pdf with mean μ and…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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LetX1,X2,...,Xn be a sequence of independent and identically distributed random variables having the Exponential(λ) distribution,λ >0, fXi(x) ={λe−λx, x >0 0, otherwise Define the random variable Y=X1+X2+···+Xn. Find E(Y),Var(Y)and the moment generating function ofY.
X1 and X2 are two discrete random variables, while the X1 random variable takes the values x1 = 1, x1 = 2 and x1 = 3, while the X2 random variable takes the values x2 = 10, x2 = 20 and x2 = 30. The combined probability mass function of the random variables X1 and X2 (pX1, X2 (x1, x2)) is given in the table below a) Find the marginal probability mass function (pX1 (X1)) of the random variable X1.b) Find the marginal probability mass function (pX2 (X2)) of the random variable X2.c) Find the expected value of the random variable X1.d) Find the expected value of the random variable X2.e) Find the variance of the random variable X1.f) Find the variance of the random variable X2.g) pX1 | X2 (x1 | x2 = 10) Find the mass function of the given conditional probability.h) pX2 | X1 (x2 | x1 = 2) Find the mass function of the given conditional probability.i) Are the random variables X1 and X2 independent? Show it. The combined probability mass function of the random variables X1 and X2 is below
LetX1,X2,...,Xn be a sequence of independent and identically distributed random variables having the Exponential(λ) distribution,λ >0, fXi(x) ={λe−λx, x >0 0, otherwise (a) Show that the moment generating function mX(s) :=E(esX) =λ/(λ−s) for s< λ;
Which of the following processes (Xt)t is weakly stationary? A: Xt = 1:6 + Xt 1 + V tB: Xt = 0:6 Xt-1 +V tC: Xt = 0:8 Xt-1 + V tD: Xt = 0:8 t + 0:6 V t – 1 The term (t) is always assumed to be white noise with variance one
1)Let x be a random variable Gaussian with zero mean and variance 1. Find:a)The conditional pdf and pdf of x given x > 0;b)E [ x| x>0 ]c)Var [ x | x >0]
13) Random variables X and Y have joint pdf fXY={4xy, 0≤x≤1, 0≤y≤1fXY={4xy, 0≤x≤1, 0≤y≤1 Find Correlation and Covariance
X 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-Y2
Consider the time series xt = β1 + β2t + wt, where β1 and β2 are known constants and wt is a white noise process with variance σ2 w. (a) Determine whether xt is stationary. (b) Show that the process yt = xt − xt−1 is stationary. (c) Show that the mean of the moving average vt = 1 2q + 1 q j=−q xt−j is β1 + β2t, and give a simplified expression for the autocovariance function.
Consider a function F (x ) = 0, if x < 0 F (x ) = 1 − e^(−x) , if x ≥ 0 Is the corresponding random variable continuous?
A stochastic process (SP) X(t) is given byX(t) = Asin(ωt + Φ)where A and Φ are independent random variables and Φ is uniformly distributed between 0 and 2π.a) Calculate mean E[X(t)]. b) Calculate the auto-correlation RX (t1,t2).c) Is X(t) wide sense stationary (WSS)? Justify your answer.Now consider that X(t) is a Gaussian SP with mean μX (t) = 0.5 and auto-correlation RX (t1,t2) =10e−14 |t1−t2|. Let Z = X(5) and W = X(9) be the two random variables.d) Calculate var(Z), var(W), and var(Z + W). e) Calculate cov(ZW).
1. Consider the Gaussian distribution N (m, σ2).(a) Show that the pdf integrates to 1.(b) Show that the mean is m and the variance is σ.
Let i_t denote the effective annual return achieved on an equity fund achieved between time (t -1) and time t. Annual log-returns on the fund, denoted by In(1 + i_t) , are assumed to form a series of independent and identically distributed Normal random variables with parameters u = 6% and o = 14%.An investor has a liability of £10,000 payable at time 15. Calculate the amount of money that should be invested now so that the probability that the investor will be unable to meet the liability as it falls due is only 5%. Using only formulas, no tables

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