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5. Which estimator is best?: A political party finances its activities through a lottery. The lottery is organized so that each participant has the opportunity to win in three independent rounds. According to the party's advertising material, there is a different chance of winning in the three rounds. If we denote the probability of winning in the first round with , the probability of winning in the second round is 20, and in the third round 50. Here is a parameter that satisfies < 1/5. In both the second and third rounds, all participants have the same chances of winning, regardless of whether they have already won a prize in an earlier round or not. We introduce the three random variables X, Y and Z that indicate whether you get a prize in each of the three rounds. We let X = 1 if the first round gives a prize, and X = 0 otherwise. Similarly for Y in the second round (Y= 1 or Y=0), and Z in the third round. (a) Show that these have expectation and variance: E(X) = 0, E(Y) = 20, E (Z) = 50, 1 §₁ = (X + Y + Z), $1 V (X) = (1 - 0) V (Y) = 20 (1 - 20) V (Z) = 50(1- 50) You will determine an estimate for the probability from the three observations X, Y, Z. You propose to use one of these estimators: - which when we evaluate for Y 82 = 1/² (x + 1/2 + 1/7). $2 X (b) Find the expectation and variance of $₁ and 2 when = 0.1. Decide which of the estimators is best (when = 0.1). Hint: We have: 1 1 1 V (§2) = 3²7 (Þ(1 − 4) +; (0(1 − ¢) + 22 26(1 − 26) +; 20(1 - = 0.1 gives V (₂) = 0.0162. − 2) + 250(1 − 50)) 1-5ø)).

5. Which estimator is best?: A political party finances its activities through a lottery. The lottery is organized so that each participant has the opportunity to win in three independent rounds. According to the party's advertising material, there is a different chance of winning in the three rounds. If we denote the probability of winning in the first round with , the probability of winning in the second round is 20, and in the third round 50. Here is a parameter that satisfies < 1/5. In both the second and third rounds, all participants have the same chances of winning, regardless of whether they have already won a prize in an earlier round or not. We introduce the three random variables X, Y and Z that indicate whether you get a prize in each of the three rounds. We let X = 1 if the first round gives a prize, and X = 0 otherwise. Similarly for Y in the second round (Y= 1 or Y=0), and Z in the third round. (a) Show that these have expectation and variance: E(X) = 0, E(Y) = 20, E (Z) = 50, 1 §₁ = (X + Y + Z), $1 V (X) = (1 - 0) V (Y) = 20 (1 - 20) V (Z) = 50(1- 50) You will determine an estimate for the probability from the three observations X, Y, Z. You propose to use one of these estimators: - which when we evaluate for Y 82 = 1/² (x + 1/2 + 1/7). $2 X (b) Find the expectation and variance of $₁ and 2 when = 0.1. Decide which of the estimators is best (when = 0.1). Hint: We have: 1 1 1 V (§2) = 3²7 (Þ(1 − 4) +; (0(1 − ¢) + 22 26(1 − 26) +; 20(1 - = 0.1 gives V (₂) = 0.0162. − 2) + 250(1 − 50)) 1-5ø)).

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
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5b)

5. Which estimator is best?: A political party finances its activities through a lottery. The lottery is organized so that each participant has the opportunity to win in three independent rounds. According to the party's advertising material, there is a different chance of winning in the three rounds. If we denote the probability of winning in the first round with p, the probability of winning in the second round is 20, and in the third round 50. Here is a parameter that satisfies < 1/5. In both the second and third rounds, all participants have the same chances of winning, regardless of whether they have already won a prize in an earlier round or not. We introduce the three random variables X, Y and Z that indicate whether you get a prize in each of the three rounds. We let X = 1 if the first round gives a prize, and X = 0 otherwise. Similarly for Y in the second round (Y = 1 or Y = 0), and Z in the third round. (a) Show that these have expectation and variance: E(X) = P, E(Y) = 20, E (Z) = 50, You will determine an estimate for the probability from the three observations X, Y, Z. You propose to use one of these estimators: Hint: We have: 1 $1 ₁=(X+Y+Z), (X + Y + Z), 8 = V (X) = (1 - 0) V (Y) = 20 (1 - 20) V (Z) = 50(150) (b) Find the expectation and variance of 1 and 2 when is best (when 0.1). which when we evaluate for 1 Y $2 = } (x + 7 + ²7 ) X 2 = 0.1 gives V (₂) = 1 V (§2) = 3 27 (Þ(1 − 4) + (1 − ¢) + ½⁄2 20(1 − 26) + 1 3² 22 4² 0.1. Decide which of the estimators = 0.0162. (5ø(1 - 5ø)).
Transcribed Image Text:5. Which estimator is best?: A political party finances its activities through a lottery. The lottery is organized so that each participant has the opportunity to win in three independent rounds. According to the party's advertising material, there is a different chance of winning in the three rounds. If we denote the probability of winning in the first round with p, the probability of winning in the second round is 20, and in the third round 50. Here is a parameter that satisfies < 1/5. In both the second and third rounds, all participants have the same chances of winning, regardless of whether they have already won a prize in an earlier round or not. We introduce the three random variables X, Y and Z that indicate whether you get a prize in each of the three rounds. We let X = 1 if the first round gives a prize, and X = 0 otherwise. Similarly for Y in the second round (Y = 1 or Y = 0), and Z in the third round. (a) Show that these have expectation and variance: E(X) = P, E(Y) = 20, E (Z) = 50, You will determine an estimate for the probability from the three observations X, Y, Z. You propose to use one of these estimators: Hint: We have: 1 $1 ₁=(X+Y+Z), (X + Y + Z), 8 = V (X) = (1 - 0) V (Y) = 20 (1 - 20) V (Z) = 50(150) (b) Find the expectation and variance of 1 and 2 when is best (when 0.1). which when we evaluate for 1 Y $2 = } (x + 7 + ²7 ) X 2 = 0.1 gives V (₂) = 1 V (§2) = 3 27 (Þ(1 − 4) + (1 − ¢) + ½⁄2 20(1 − 26) + 1 3² 22 4² 0.1. Decide which of the estimators = 0.0162. (5ø(1 - 5ø)).
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