Question

A discrete random variable X follows the geometric distribution with parameter p, written X ∼ Geom(p), if its distribution function is

A discrete random variable X follows the geometric distribution with parameter p, written X Geom(p), if its distribution function is 1x(z) = p(1-P)"-1, ze(1, 2, 3, ). The Geometric distribution is used to model the number of flips needed before a coin with probability p of showing Heads actually shows Heads. a) Show that fx(x) is indeed a probability mass function, i.e., the sum over all possible values of r is one. b) Let lc< 1, and consider the sum S=1+2c + 3c2 + 4c3 + By multiplying S by c, we find: cS- c+2c2 +3c3 +4c4 +... By combining these two expressions, prove that S-1/(1 - c)2 c) Prove that the expected value of X Geom(p) is 1/p. Hint: the result of the previous problem might be useful d) Let xi, ,Zn be n observations of a geometric random variable with parameter p. Prove that the maximum likelihood estimator for the parameter p is (1 Σ.1m)-1.

Answer 1

$(a)~\sum_{x=1}^\infty p(1-p)^{x-1}=p(1&plus;(1-p)&plus;(1-p)^2&plus;....)\\ =p\times \frac{1}{1-(1-p)},~since~0<p<1\\ =1\\ (b)~S=1&plus;2c&plus;3c^2&plus;4c^3&plus;...\\ cS=c&plus;2c^2&plus;3c^3&plus;4c^4&plus;...\\ (S-cS)=(1-c)S=1&plus;c&plus;c^2&plus;c^3&plus;...=\frac{1}{1-c}~since~|c|<1\\ or,~S=\frac{1}{(1-c)^2}.\\ (c)~E(X)=\sum_{x=1}^\infty xp(1-p)^{x-1}=p(1&plus;2(1-p)&plus;3(1-p)^2&plus;4(1-p)^3&plus;...)\\ =p\times \frac{1}{(1-(1-p))^2}~using~the~result~of~(b)~and~put~c=1-p\\ =\frac{1}{p}.\\ (d)~The~likelihood~function~of~p=L(p)=p^{n}(1-p)^{\sum_{i=1}^nx_i-n}\\ l(p)=\log(L(p))=n\log p&plus;(\sum_{i=1}^nx_i-n)\log (1-p)\\ \frac{\mathrm{d} l(p)}{\mathrm{d} p}=\frac{n}{p}-\frac{\sum_{i=1}^nx_i-n}{1-p}=0\\$

$\Rightarrow n-p\sum_{i=1}^nx_i=0\Rightarrow \hat{p}=\frac{1}{\bar{x}},~where,~\bar{x}=\frac{1}{n}\sum_{i=1}^nx_i\\ \frac{\mathrm{d}^2 l(p)}{\mathrm{d} p^2}=-\frac{n}{p^2}-\frac{\sum_{i=1}^nx_i-n}{(1-p)^2}<0~for~p=\hat{p}.\\ Hence~MLE~of~p=\frac{1}{\bar{x}}=(\bar{x})^{-1}=\left(\frac{1}{n}\sum_{i=1}^nx_i \right )^{-1}.$