This problem explores the question of testingHo p= 0.5 versus Ho p < 0.5,based on a random sample X₁, X2, ,, Xn from the Geometric (p) distribution.Recall that there is an ambiguity surrounding the definition of the Geometric (p)distribution, so let us unequivocally specify which distribution we are going to workwith. In the experiment of repeatedly tossing a coin with the probability of getting a Headin a single toss of the coin equal to p = (0, 1) till a Head is obtained, let X denote thenumber of Tails obtained before the experiment is terminated. Then X has theGeometric (p) distribution.Further recall that X₁, X2,, Xn is a random sample of size n from the Geometric (p)distribution means X = (X₁, X2,, Xn) is a collection of mutually independentrandom variables, with each X, having the Geometric (p) distribution. Before proceedingfurther, let us recall the following definition.Definition A collection of random variables {Y₁, Y2,..., Yr} defined on the same samplespace, with S, denoting the support of Y, for 1 ≤ i ≤r, is said to be mutuallyindependent if, \ (81, 82, ..., sr) €, S₁, the Cartesian product of (S₁, S2, · · ·, Sr),P(₁₂(Y₁ = $i)) = [[P(Y₁ = $₁).1=1Remember that if {Y₁, Y2,, Yr} is a collection of mutually independent randomvariables, then for any 1 ≤ m ≤ r, the random variablesU = g(Y₁, Y2,, Ym) and V = h(Ym+1, Ym+2,, Y₂)are independent. (a) Find the probability mass function of (X₁ + X₂).n(b) Using the result of (a), guess what the probability mass function of EX, will be andi=1substantiate your guess by using the method of induction.In (c), (d), (e), and (f), consider the problem of testingat a = 0.038406.Ho:p0.5 versus Ha: p = 0.2(c) Write down the likelihood function L(p; x), where x = (x1, x2,,xn) is the datasample realized from the random sample X = (X1, X2,..., Xn).points.(d) Write down the expression for the Neyman-Pearson test statistic T(x) and thecorresponding rejection region T as outlined in the posted notes on Neyman-PearsonLemma.

The random variable which takes an integer value is called a discrete random variable.The geometric…

(a) Find the probability mass function of (X₁ + X₂). (b) Using the result of (a), guess what the probability mass function of X, will be and i=1 substantiate your guess by using the method of induction. In (c), (d), (e), and (f), consider the problem of testing Ho p = 0.5 versus Ha: p = 0.2 at a = 0.038406. n (c) Write down the likelihood function L(p; x), where x = (x1, x2,,xn) is the data sample realized from the random sample X = (X₁, X2,..., Xn). points. (d) Write down the expression for the Neyman-Pearson test statistic T(x) and the corresponding rejection region T as outlined in the posted notes on Neyman-Pearson Lemma.

(a) Find the probability mass function of (X₁ + X₂). (b) Using the result of (a), guess what the probability mass function of X, will be and i=1 substantiate your guess by using the method of induction. In (c), (d), (e), and (f), consider the problem of testing Ho p = 0.5 versus Ha: p = 0.2 at a = 0.038406. n (c) Write down the likelihood function L(p; x), where x = (x1, x2,,xn) is the data sample realized from the random sample X = (X₁, X2,..., Xn). points. (d) Write down the expression for the Neyman-Pearson test statistic T(x) and the corresponding rejection region T as outlined in the posted notes on Neyman-Pearson Lemma.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter2: Equations And Inequalities

Section2.7: More On Inequalities

Problem 44E

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