A party finance themselves with a lottery. It is organized that every partecipant has a possibility to win in three independent occasions. There are different chances to win every occasion. The first time the probability to win is θ, the second occasion it is 2θ, and the last chance is 5θ. θ is a parameter that satisfifyes θ < max(1/2, 1/5). The three random variables X1, X2, X3 are equal to 1 if you win on that occasion and zero otherwise. j=1, 2, 3 a) What are expected values and variance of the X's b) Given the estimators: θhat1 = 1/8(X1+X2+X3) and θhat2 =1/3 (X1 + X2/2 + X3/5) Show that the estimators are unbiased. Find the function of the estimators expressed with θ. Which of them gives the best result when θ =0.2?
A party finance themselves with a lottery. It is organized that every partecipant has a possibility to win in three independent occasions. There are different chances to win every occasion. The first time the probability to win is θ, the second occasion it is 2θ, and the last chance is 5θ. θ is a parameter that satisfifyes θ < max(1/2, 1/5).
The three random variables X1, X2, X3 are equal to 1 if you win on that occasion and zero otherwise. j=1, 2, 3
a) What are
b) Given the estimators:
θhat1 = 1/8(X1+X2+X3) and θhat2 =1/3 (X1 + X2/2 + X3/5)
Show that the estimators are unbiased. Find the
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