I not only gave this lecture today I work with order stats.

so far I have ...

Here it is....

Forget about pdf's, you can differentiate the cdf if you wish.

F(x)=1-exp{-(x-theta)} if x exceeds theta and zero otherwise.

So the cdf of Y_n is (I'm using < since it's continuous and I don't want to write <=)

G(y)=P(Y_n<y) = 1-P(Y_n>y)=1-[1-F(y)]^n

=1- exp{-n(x-theta)}

THUS

P(|Y_n-theta|>espislon)=P(theta-epsilon<Y_n<theta+epsilon)

=P(theta<Y_n<theta+epsilon) due to the support

=G(theta+epsilon)-G(theta)

=(1- exp{-n(theta+epsilon-theta)})-(1- exp{-n(theta-theta)})

=1-exp{-n epsilon} -1 +1

=1-exp{-n epsilon}

->1

for all epsilon as n-> infinity