Question

Let X1, ..., Xn be a random sample from a geometric distribution, where each Xi is counting the number of failures until the first success in a sequence of independent Bernoulli trials with constant probability p of success. (a) Find the distribution of Sn = ?i=1 to n Xi. (b) Find E(Sn) and Var(Sn). (c) Let Xn_bar = 1/n Sn. Find E(Xn_bar) and Var(Xn_bar). (d) Find a consistent estimator of p.

          Let X1, ..., Xn be a random sample from a geometric distribution, where each Xi is counting the number of failures until the first success in a sequence of independent Bernoulli trials with constant probability p of success.

(a) Find the distribution of
Sn = ?i=1 to n Xi.

(b) Find E(Sn) and
Var(Sn).

(c) Let Xn_bar = 1/n Sn. Find
E(Xn_bar) and Var(Xn_bar).

(d) Find a consistent
estimator of p.

Let X1, ..., Xn be a random sample from a geometric distribution, where each Xi is counting the number of failures until the first success in a sequence of independent Bernoulli trials with constant probability p of success.

(a) Find the distribution of
Sn = ?i=1 to n Xi.

(b) Find E(Sn) and
Var(Sn).

(c) Let Xnbar = 1/n Sn. Find
E(Xnbar) and Var(Xnbar).

(d) Find a consistent
estimator of p.

Added by Alejandra H.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Madhur L

07/26/2023

Step 1

Now, let $S_n = \sum_{i=1}^n X_i$. Since $X_i$ are independent and identically distributed (i.i.d.) geometric random variables, the distribution of $S_n$ is the sum of $n$ independent geometric distributions. This is known as the negative binomial distribution Show more…

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Let X1, ..., Xn be a random sample from a geometric distribution, where each Xi is counting the number of failures until the first success in a sequence of independent Bernoulli trials with constant probability p of success. (a) Find the distribution of Sn = ∑i=1 to n Xi. (b) Find E(Sn) and Var(Sn). (c) Let Xn_bar = 1/n Sn. Find E(Xn_bar) and Var(Xn_bar). (d) Find a consistent estimator of p.