Covariance

Definition (Covariance).

The Covariance between and is

Remark.

If is independent
The converse may not be true

Exercise.

(i)
(ii)
(iii)
(iv)

Proof.
(i)
(ii)
(iii)

(iv) let

Lemma.

Proof.

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Definition (Sample mean and Sample Variance).

Definition (Independence of Random Variable ).

let are random variable they are independent if

Definition (identical Random Variable ).

let are random variable they are Identical if

let are independent and identically distributed random variable
let

lets find and

Find ( is the sample variance )

Lets separately find and

Basically fuck the above as its too lengthy and do it in a better way

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Example (Sampling from finite population ).

Polulation of people voting in election is a candiditte for election

for each person in population

Proportion in favor of
Take sample of size from population.
Take average of for sample is estimator for V

let be the number of people in flavor of in the sample

is the proportion of people in sample who favor

conform the above stuff

We second step ^02f340 (2.1)