2- a) Find the value of k for which this is a valid joint PDF for the random variables X and Y.f(x,y) = {k(x² + y²), 0≤x≤1 and 0 ≤ y ≤10,otherwiseb) Compute the marginal PDF's.c) Determine whether the random variables are independent or not.d) Compute the expected value and the variance of X

The joint PDF of X and Y is:f(x,y)=k(x2+y2)0≤x≤1 and 0≤y≤1Objective : a) Find Kb) Marginal…

Answered: 2- a) Find the value of k for which…

A First Course in Probability (10th Edition)

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Chapter1: Combinatorial Analysis

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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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Let the random variable X be defined on the support set (1,2) with pdf fX(x) = (4/15)x3, Find the variance of X.
Consider the one-way analysis of variance model Xij = µ + a; + Eij, i= 1,.., m, j= 1,..., ni, where ɛij ~ N(0,0²) are independent. Let n= n1 + · ·+ nm, ... ni 1 ni 1 X;. >Xii for i = 1,..., m, and X. = - ΣΣΧ. ni j=1 i=1 j=1 (a) Show that SS(TO) = SS(T) + SS(E), where SS(TO) = E E i=12j=1 (Xij – X..)², m m ni SS(T) Σι (X.-Χ.) and SS(E) -ΣΣ (Χ- X.) . i=1 i=1 j=1
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Suppose that X, Y and Z are independent. The expected value is given by E(X) = 0, E(Y) = 1, E(Z) = 5 We also know that the variance is given by Var(X)=1, Var(Y)=1, Var(Z)=25 The values of the covariance are Cov(X,Y)=2, Cov(Y,Z)=1 Which of the satements are true? 1. Cov(ex,Y2 + 1) = 0 2. Е(XY)-1 3. If a and b are real numbers then Cov(aX+bZ, Y+Z)=2a+25b 4. Cov(X,Y)=0
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Exercise 39 Let X1 ~ N(µ1,07) and X2 ~ N(u2, o3). Find the expected value and variance of Y = X1+ X2.
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Suppose we collect n samplesX1,X2, . . . ,Xn from a uniform distribution U[θ,1]. We know the maximum likelihood estimator of θ is θ = min(Xi). (1) What is the CDF of θ? (2) What is the pdf of θ? (3) Find the bias of θ. (4) Find the variance of θ. (5) Find the MSE of θ.
2) Let X₁, X2, ..., Xn be a random sample from the pdf f(x;0) = 0xª−¹, 0≤x≤ 1,0 <0 < ∞. Find the MLE of 0 and show that its variance approaches 0 as n approaches ∞o.
Page 264, 4.4.12
Let Y < Y, < Y3 be the order statistic of a random sample of size 3 from the uniform distribution having pdff (x;0) = 1/0,0

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