# Math Help - consitent estimator

1. ## consitent estimator

let Xn , n= 1, 2, . . . . . be a sequence of r.v st

p(Xn = a) = 1 - (1/n), a in R

p(Xn = n^k) = 1/n k>1 , k fixed

show Xn is consitent a consistant estimator for a.

i.e show for all e>0 p(|Xn - a|>= e) tends to 0 as n tends to infinity

any idears

thanks for you help

2. The probability of that event is 1/n, which clearly goes to zero...

$P(|X_n-a|>\epsilon)\le P(X_n\ne a)=1/n\to 0$

3. i kinda see i dont really understand why the first inequality is true though

4. Look at the number line....

$(-\infty,a-\epsilon)\cup(a+\epsilon,\infty)$

is contained in the interval

$(-\infty,a)\cup(a,\infty)$

Hence it has smaller probability.

5. ok i see thank you