Question

1. Let Y be a Bernoulli random variable with success probability P(Y = 1) = p. Let Y_1, Y_2, ..., Y_n be and i.i.d sample from this distribution. Let p? = (1/n) ?_{i=1}^{n} 1{Y_i = 1} . (a) Show that p? is an unbiased estimator of p. (b) Show that Var[p?] = p(1-p)/n .

          1. Let Y be a Bernoulli random variable with success probability P(Y = 1) = p. Let Y_1, Y_2, ..., Y_n be and i.i.d sample from this distribution. Let

p? = (1/n) ?_{i=1}^{n} 1{Y_i = 1} .

(a) Show that p? is an unbiased estimator of p.
(b) Show that Var[p?] = p(1-p)/n .

1. Let Y be a Bernoulli random variable with success probability P(Y = 1) = p. Let Y1, Y2, ..., Yn be and i.i.d sample from this distribution. Let

p? = (1/n) ?i=1^n 1Yi = 1 .

(a) Show that p? is an unbiased estimator of p.
(b) Show that Var[p?] = p(1-p)/n .

Added by Tom-S J.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Sri K

10/08/2023

Step 1

i.d, we can rewrite the expectation as: $$ E[\hat{p}] = \frac{1}{n}\sum_{i=1}^n E[Y_i] $$ We know that $E[Y_i] = p$ for a Bernoulli random variable, so: $$ E[\hat{p}] = \frac{1}{n}\sum_{i=1}^n p = \frac{1}{n} \cdot np = p $$ Thus, $\hat{p}$ is an unbiased Show more…

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Let Y be a Bernoulli random variable with success probability P(Y = 1) = p. Let Y1, Y2, Yn be an i.i.d sample from this distribution. Let p = (1/n) * Σ(Yi = 1) for i=1 to n Show that p is an unbiased estimator of p. Show that Var[p] = p(1-p)/n