9.19 Let X and Y be two continuous random variables, with joint proba- bility density function f(x, y): - 30 -50x²-50y² +80xy for - 9.19 Let X and Y be two continuous random variables, with joint proba-bility density functionf(x, y) =30-502²-50y²+80zyㅠfor -∞ < x < ∞ and - < y < x; see also Figure 9.2.134 9 Joint distributions and independencea. Determine positive numbers a, b, and c such that50x² 80xy +50y² = (ay-bx)² + cx².b. Setting u = , and olo, show that=and use this to show that(√50y-√32.x)² -1 (³-7") ²2Le e-(√50y-√32x)² dy=√2π10c. Use the results from b to determine the probability density function fxof X. What kind of distribution does X have?

Answered: Image /qna-images/answer/e09e1960-d5f0-45fb-b1a1-5c90c893660c.jpg

Answered: 9.19 Let X and Y be two continuous…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Let X1,X2,... be a sequence of identically distributed random variables with E|X1|<∞ and let Yn = n−1max1≤i≤n|Xi|. Show that limnE(Yn) = 0
10 Suppose that X is a continuous random variable with pdf given by X(x) =cxα−1e−(x/β)α for x≥0, where α >0,β >0 are some parameters of the distribution, and c is a constant. (a) Find the value of c such that fX is a valid pdf.(b) Find the cdf and the quantile function of X.(c) Ifα= 2 andβ= 4, calculate P(X <1) and find the quartiles of X.
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
4 (b) An insurance company provides customers with both auto and home insurance policies. For a particular customer, Χ is the deduction on his or her auto policy and Y is the deduction on the home policy. Possible values of Χ are K100 and K250, and for Y are K0, K100 and K200. The joint probability density function for ( ) ,YΧ is given by the following table: Χ Y K100 K250 K0 0.20 0.05 K100 0.10 0.15 K200 0.20 0.30 iv. If we look only at those insurance customers selecting the lowest auto mobile insurance deduction (K100), what is the probability that a randomly selected\ customer will also select the lowest home deduction (K0). v. Compute the correlation coefficient of Χ and Y
Let X1, ...., Xn be a random sample from a population with θ unknown and given by the density f(x; θ) = ( 1 2θ √2 x e − √2 x θ if x > 0 0 if x ≤ 0 1. Show that E(X) = 2θ 2 and E( √2 X) = θ (Hint: you may use that R ∞ 0 e −z z α−1dz = (α − 1)! for every α ∈ N). 2. Show that the statistic θbn := 1 n Xn i=1 p2 Xi (1) is an unbiased estimator of θ. 3. Give the definition of a consistent estimator. 4. Show that the estimator θbn given in relation (1) is a consistent estimator of θ. 5. Show that the estimator θbn is a minimum variance estimator of θ. (Hint: use the Cramer-Rao inequality given by var(θb) ≥ 1 nE ∂ ln(f(X;θ) ∂θ 2
Let 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!
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?
5. Suppose Z is a Gaussian random variable with mean 0 and 1. (a) Find the moment generating function of Z2. (b) Find the density function of Z2.
Let X be a Gaussian random variable (0,1). Let M = ln(5*X) be a derived random variable. What is E[M]?
Let X1, X2, X3, . . . be a sequence of independent Poisson distributed random variables with parameter 1. For n ≥ 1 let Sn = X1 + · · · + Xn. (a) Show that GXi(s) = es−1.(b) Deduce from part (a) that GSn(s) = ens−n.
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.
11) Consider the following joint probability mass function (PMF): f(X,Y)= kxy2 s= {(1,1), (1,2), (2,2)} in which the support (sample space) is Find k. A. k=1/3. B. k=1/7. C. k=1/13. D. k=1. E. None of the above.

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