(c) Find A, the maximum likelihood estimator for A. (You do not need to prove that your solutionis a max).(d) Show that your estimator X from (c) is an unbiased estimator of X. Let X1, X2, X3, and X4 be independent Poisson random variables, with parametersA₁ = 3A, A2 = A, A3 = 9A, and A4 = 4A, respectively.(a) Find and simplify the joint pmf f(x1, x2, x3, x4; A1, A2, A3, A4).(b) Find L(A) and (A), the likelihood and log-likelihood functions.

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Answered: Let X1, X2, X3, and X4 be independent…

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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Exercise 3. Let X be a random variable with mean µ and variance o². For a € R, consider the expectation E((X − a)²). a) Write E((X - a)²) in terms of a, μ and σ². b) For which value a is E((X − a)²) minimal? c) For the value a from part (b), what is E((X − a)²)?
Suppose we roll 3 fair dice and let X be the number of twos we roll.(a) Find the probability mass function of X.(b) Find the expected value of X.(c) Find the variance of X.(d) Find the probability-generating function of X.(e) Find the moment-generating function of X
A random variable X has a N(0,1) distribution. Use the moment generating function of X to compute (a) the third and (b) fourth moments of X, i.e., E(X³) and E(X4).
Exercise 3. Let X be a random variable with mean and variance o². For a € R, consider the expectation E((X-a)²). a) Write E((X-a)²) in terms of a, u and o². b) For which value a is E((X - a)²) minimal? c) For the value a from part (b), what is E((X-a)²)?
Let X be a random variable with pmf P(O)=0.2, P(1)-0.5, P(2)=0.3. Let F(X) be the cdf of X. (a) Find P(X<0.7) (b) Find F(1.3). (c) Find F(3).
Identify the distributions of the random variables with the following moment- generating functions and derive the mean and the variance: a) My(t) = (1 – 4t)-2 b) My(t) = e-5t+6t² %3D
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Consider an AR(2) model: STEPS; -Write the equation of the AR(2) model- Determine the model based on the delay operator.- Find the expected value of the model- Find the variance of the model.- Find the covariances associated with 1,2, and s steps.- Find the associated correlation indices of 1,2, and s steps.- Assume that information is available up to time $h$ and the function that contains the accumulated information f(h, h-1,……), determine the forecasts and the error associated with 1,2, and s steps.- Assume that there is an initial value of the returns, denoted by $r_0$, and we have the AR(2) model at time rt, convert the AR(2) model into its equivalent MA(t)
Let X11 X12, Xini and X21, X22, X2n2 be two independent random samples of size n₁ and n₂ from two normal populations N(₁, 2) and N(2, 2) respectively. (a) Derive the maximum likelihood estimators (mle's) of all the parameters in the first population (X₁). Using analogy, state the mle's of the parameters of the second population. (b) Find the pooled estimator of the common variance when it is assumed that of = = ². Suggest an unbiased estimator of o². |

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Let X1, X2, X3, and X4 be independent Poisson random variables, with parameters A₁ = 3A, A2 = A, A3 = 9A, and A4 = 4A, respectively. (a) Find and simplify the joint pmf f(x1, x2, x3, x4; A1, A2, A3, A4). (b) Find L(A) and (A), the likelihood and log-likelihood functions.