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2 A discrete random variable Y takes the values -1, 0 and 1 with probabilities 10, 1-0 and 10 respectively. Let Y₁ and Y₂ be two independent random variables, each with the same distribution as Y. (a) List the possible values of {Y₁, Y₂) that may arise and calculate the probability of each. Verify that your probabilities sum to unity. (b) By calculating the value of (Y₂ - Y₁)² for each possible pair (Y₁, Y₂), determine the sampling distribution of (Y₂ - Y₁)². ... (c) Show that X = (Y2Y₁)² is an unbiased estimator for 0, and find Var(X) as a function - of 0. (d) Now suppose that Y1, Y2..... Y, are n independent observations on the random variable Y. Since 0 is the probability that Y is not zero, a possible estimator for 0 is the proportion, Z, of non-zero values among Y₁, Y2,..., Y. Write down the mean and variance of Z, and state, giving your reasons, which of X and Z you would prefer as an estimator for 8 when n = 2.

2 A discrete random variable Y takes the values -1, 0 and 1 with probabilities 10, 1-0 and 10 respectively. Let Y₁ and Y₂ be two independent random variables, each with the same distribution as Y. (a) List the possible values of {Y₁, Y₂) that may arise and calculate the probability of each. Verify that your probabilities sum to unity. (b) By calculating the value of (Y₂ - Y₁)² for each possible pair (Y₁, Y₂), determine the sampling distribution of (Y₂ - Y₁)². ... (c) Show that X = (Y2Y₁)² is an unbiased estimator for 0, and find Var(X) as a function - of 0. (d) Now suppose that Y1, Y2..... Y, are n independent observations on the random variable Y. Since 0 is the probability that Y is not zero, a possible estimator for 0 is the proportion, Z, of non-zero values among Y₁, Y2,..., Y. Write down the mean and variance of Z, and state, giving your reasons, which of X and Z you would prefer as an estimator for 8 when n = 2.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.8: Probability
Problem 6E
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2 A discrete random variable Y takes the values -1, 0 and 1 with probabilities 10, 1-0 and 10 respectively. Let Y₁ and Y₂ be two independent random variables, each with the same distribution as Y. (a) List the possible values of {Y1, Y2) that may arise and calculate the probability of each. Verify that your probabilities sum to unity. (b) By calculating the value of (Y₂ - Y₁)² for each possible pair (Y₁, Y2), determine the sampling distribution of (Y₂ - Y₁)².. (c) Show that X = (Y₂Y₁)² is an unbiased estimator for 0, and find Var(X) as a function of 0. (d) Now suppose that Y₁, Y2..... Y, are n independent observations on the random variable Y. Since is the probability that Y is not zero, a possible estimator for 0 is the proportion, Z, of non-zero values among Y₁, Y2,, Y. Write down the mean and variance of Z, and state, giving your reasons, which of X and Z you would prefer as an estimator for 8 when n = 2.
Transcribed Image Text:2 A discrete random variable Y takes the values -1, 0 and 1 with probabilities 10, 1-0 and 10 respectively. Let Y₁ and Y₂ be two independent random variables, each with the same distribution as Y. (a) List the possible values of {Y1, Y2) that may arise and calculate the probability of each. Verify that your probabilities sum to unity. (b) By calculating the value of (Y₂ - Y₁)² for each possible pair (Y₁, Y2), determine the sampling distribution of (Y₂ - Y₁)².. (c) Show that X = (Y₂Y₁)² is an unbiased estimator for 0, and find Var(X) as a function of 0. (d) Now suppose that Y₁, Y2..... Y, are n independent observations on the random variable Y. Since is the probability that Y is not zero, a possible estimator for 0 is the proportion, Z, of non-zero values among Y₁, Y2,, Y. Write down the mean and variance of Z, and state, giving your reasons, which of X and Z you would prefer as an estimator for 8 when n = 2.
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