Let X1,...,Xn be a random sample from a geometric distribution with the probability mass
function:
f(x,p)= p(1-p)x x=0,1,2,3...
And the expected value:
E(X)=(1-p)/p
For example, the lifetime of a light bulb can be modeled as a geometrically distributed random
variable with p being the probability of bulb burning out in one trial (switch).
Construct a method of moment’s estimator of p.
Solution
If E(x) = (1-p)/p, cross-multiplying, pE(x) = 1 - p pE(x) + p = 1 p(1+E(x)) = 1 p =
1/(1+E(x)).
Let X1,...,Xn be a random sample from a geometric distribution with .pdf
1. Let X1,...,Xn be a random sample from a geometric distribution with the probability mass
function:
f(x,p)= p(1-p)x x=0,1,2,3...
And the expected value:
E(X)=(1-p)/p
For example, the lifetime of a light bulb can be modeled as a geometrically distributed random
variable with p being the probability of bulb burning out in one trial (switch).
Construct a method of moment’s estimator of p.
Solution
If E(x) = (1-p)/p, cross-multiplying, pE(x) = 1 - p pE(x) + p = 1 p(1+E(x)) = 1 p =
1/(1+E(x))
