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The random variable X has a Bernoulli distribution with parameter p. A random sample X1, X2, . . . , Xn of size n is taken of X. Show that the sample proportion X1 + X2 + · · · + Xn n is a minimum variance unbiased estimator of p.

The random variable X has a Bernoulli distribution with parameter p. A random sample X1, X2, . . . , Xn of size n is taken of X. Show that the sample proportion X1 + X2 + · · · + Xn n is a minimum variance unbiased estimator of p.

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter13: Probability And Calculus
Section13.2: Expected Value And Variance Of Continuous Random Variables
Problem 10E
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The random variable X has a Bernoulli distribution with parameter p. A random sample
X1, X2, . . . , Xn of size n is taken of X. Show that the sample proportion
X1 + X2 + · · · + Xn
n
is a minimum variance unbiased estimator of p.
Given that X1, X2, . . . , Xn forms a random sample of size n from a geometric population with
parameter p, show that
Y =
n∑
j=1

The random variable X has a Bernoulli distribution with parameter p. A random sample X1, X2,..., Xn of size n is taken of X. Show that the sample proportion + Xn X₁ + X₂ + n is a minimum variance unbiased estimator of p. Given that X₁, X2,..., Xn forms a random sample of size n from a geometric population with parameter p, show that is a sufficient estimator of p. TL Y=Xj j=1
Transcribed Image Text:The random variable X has a Bernoulli distribution with parameter p. A random sample X1, X2,..., Xn of size n is taken of X. Show that the sample proportion + Xn X₁ + X₂ + n is a minimum variance unbiased estimator of p. Given that X₁, X2,..., Xn forms a random sample of size n from a geometric population with parameter p, show that is a sufficient estimator of p. TL Y=Xj j=1
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Calculus For The Life Sciences
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ISBN:
9780321964038
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GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
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