Sample distribution and expected value.

[kidsasd987]

Consider a scenario where samples are randomly selected with replacement. Suppose that the population has a probability distribution with mean µ and variance σ 2 . Each sample Xi , i = 1, 2, . . . , n will then have the same probability distribution with mean µ and variance σ 2 . Now, let us calculate the mean and variance of X_bar: E(X_bar) = 1/n*(E(X1) + E(X2) + · · · + E(Xn)) = 1/n (µ + µ + · · · + µ ) = µ

*X_i is independent random variable.Hello. I wonder why the expected values of Xi are the same as population average µ.

 

[BvU]

Hi,

Not sure what you mean with the probability distribution of a single sample. What's that ?

 

[kidsasd987]

Hi,

Not sure what you mean with the probability distribution of a single sample. What's that ?

I guess it means that random variable has the same probability for P(X=x), like Bernoulli random variable.


Please refer to the link above.

 

[BvU]

It's probably more like a short form of saying that the set of all possible individual x_i has the same probability distribution as ... (because it's the same population).

why the expected values of Xi are the same as population average µ

Well, that is because the expression in the definition of $\mu$ and the expression for the expectation value are identical.

 

[kidsasd987]

It's probably more like a short form of saying that the set of all possible individual x_i has the same probability distribution as ... (because it's the same population).

Well, that is because the expression in the definition of $\mu$ and the expression for the expectation value are identical.

I am sorry. Maybe I am too dumb to understand at once. Can you help me to figure out the questions below?
(*they are not homework questions but I wrote them in statement form because It'd be easier to answer.)1. Xi are the samples with n size.
Does that mean X1 can have n number of data within it? For example, let's say our population has a data set {1,2,3,4,5,6,7,8,9,10}
and X1 has a size of 2, then {1,2},{1,4},... on can be the sample X1.

2. (if 1 is correct) I understand why E(X)=μ, but how their samples E(X1),E(X2).. and on equal to μ.
E(X)=sigma(P(X=xi)*xi)
E(X1)=sigma(P(X1=xj)*xj) but the sum will be significantly smaller than E(X)?

Thanks.

 

[Amith2006]

1. Xi are the samples with n size.
Does that mean X1 can have n number of data within it? For example, let's say our population has a data set {1,2,3,4,5,6,7,8,9,10}
and X1 has a size of 2, then {1,2},{1,4},... on can be the sample X1.

2. (if 1 is correct) I understand why E(X)=μ, but how their samples E(X1),E(X2).. and on equal to μ.
E(X)=sigma(P(X=xi)*xi)
E(X1)=sigma(P(X1=xj)*xj) but the sum will be significantly smaller than E(X)?
Thanks.

$X_i$ is not a sample. It is a random variable. We find the expectation value of that random variable defined as,
$E(X_i) = \Sigma{x_iP(x_i)} = \mu$
Hope this helps!