Suppose the random variable, X, follows a Bernoulli distribution with parameter 0.The probability function of X is given by{[ 0² (1 - 0)¹-² if x = 0,1,otherwise.p(x; 0) =Let X₁, X₂, X be a random sample of size n from the population of X. Assumethat the X's are independent and identically distributed with the same parameter.(a) Based on the joint probability function of the sample, state the likelihoodfunction of the parameter. Discuss the difference between the joint probabilityfunction and the likelihood function.(b) Show that the sample mean is the maximum likelihood estimator (mle) of 0.(c) Find the method of moments estimator (mme) of 0.

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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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Suppose that the unknown X is uniformly distributed between 0 and 2. What is the expected value of (4X3 + 1)?
IT HAS BEEN FOUND THAT A FLUCTUATING ELECTRIC CURRENT X AT ONE OF THE ESKOM SUBSTATIONS HAS A UNIFORM PROBABILITY DISTRIBUTION, f(x) = a≤x≤b otherwise a+b (b-a)² WITH E(X) = AND Var (X) = 12 AS THE STATISTICIAN AT ESKOM AND HAVE BEEN GIVEN A RANDOM SAMPLE OF 7 MEASUREMENTS X₁, X₂,,X, OF THE ELECTRIC CURRENT TO INFER ABOUT THE MEAN FLUCTUATION OVER A SPECIFIC PERIOD. AFTER CAREFUL CONSIDERATION YOU HAVE IDENTIFIED THREE POSSIBLE ESTIMATORS 0₁ = X₁+X7, 0₂ = X₁ +2X₂+X AND 3 = X 3 Show which of the estimators possess this property
If the joint probability distribution of X and Y isgiven byf(x, y) = 130 (x + y) for x = 0, 1, 2, 3; y = 0, 1, 2 construct a table showing the values of the joint distribu-tion function of the two random variables at the 12 points (0, 0),(0, 1), ... ,(3, 2).
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Suppose the distribution of some random variable X is modeled as Negative binomial with parameters, p and r, where p is the probability of success and r is the total number of success over the trials. Find the method of moment (MM) estimators for (p,r), in closed algebraic form expressions based on a random sample of size n, X1, X2,...,Xn.
Suppose that the Cumulative distribution function (CDF) of the random variable X with parameter 0 > 0 is defined by F(x; θ) - 1-el x > 0 i) What is the expected value of X,E(X)? ii) What is the variance of X,V(X)?
An ordinary (fair) coin is tossed 3 times. Outcomes are thus triple of “heads” (h) and tails (t) which we write hth, ttt, etc. For each outcome, let R be the random variable counting the number of tails in each outcome. For example, if the outcome is hht, then R (hht)=1. Suppose that the random variable X is defined in terms of R as follows X=6R-2R^2-1. The values of X are given in the table below. A) Calculate the values of the probability distribution function of X, i.e. the function Px. First, fill in the first row with the values X. Then fill in the appropriate probability in the second row.
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Suppose that the Cumulative distribution function (CDF) of the random variable X with parameter 0 > 0 is defined by F(x;0) = 1 - e , x > 0 i) What is the expected value of X,E(X)? ii) What is the variance of X,V (X)?

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