Consider 10 i.i.d. random variables
𝑋
1
,
𝑋
2
,
.
.
.
,
𝑋
10
, each with expected value 0 and variance 5. What is the variance of the sample mean? That is, what is
𝑉
𝑎
𝑟
(
𝑋
¯
10
)
?

Question

Consider 10 i.i.d. random variables 
𝑋
1
,
𝑋
2
,
.
.
.
,
𝑋
10
, each with expected value 0 and variance 5.  What is the variance of the sample mean?  That is, what is 
𝑉
𝑎
𝑟
(
𝑋
¯
10
)
?

BlackTom AI · Accepted Answer

For i.i.d. random variables the variance of their average shrinks by the sample size: Var(X̄n) = σ^2 / n. This key result follows from linearity of variance for independent variables.

1) Relevant concepts:
- For independent random variables, Var(aY) = a^2 Var(Y) and Var(∑ Yi) = ∑ Var(Yi).
- If Xi are i.i.d. with Var(Xi) = σ^2, then Var(X̄n) = Var((1/n)∑ Xi) = σ^2/n.
- Independence is crucial; otherwise covariance terms appear.

2) Step-by-step reasoning:
We want Var(X̄10) where X̄10 = (1/10) ∑_{i=1}^{10} Xi and Var(Xi) = 5.
Apply scaling and sum rules for variance.

3) Calculations with intermediate steps:
Var(X̄10) = Var((1/10) ∑_{i=1}^{10} Xi)
= (1/10)^2 Var(∑_{i=1}^{10} Xi)            (pull out constant squared)
= (1/100) ∑_{i=1}^{10} Var(Xi)            (independence lets us sum variances)
= (1/100) × (10 × 5)                      (each Var(Xi) = 5)
= (1/100) × 50
= 50/100
= 0.5.

Alternatively, directly using σ^2/n: 5/10 = 0.5.

4) Why the correct answer is correct:
Because averaging 10 independent observations each with variance 5 reduces the variance by a factor of 10. The algebra above shows the exact reduction from 5 to 0.5.

5) Why the other choices are incorrect:
- 0.05 would be 5/100; this arises if one mistakenly divides by n^2 instead of n (i.e., (σ^2)/(n^2)), which is wrong because the (1/n)^2 scaling applies to the sum’s variance but the sum contributes an extra factor n.
- 5 is the variance of a single Xi, not of the average; averaging reduces variance.
- 50 equals 10×5, the variance of the sum ∑ Xi, not of the sample mean (it misses the (1/10)^2 scaling).

6) Conclusion:
The variance of the sample mean is 0.5, which matches the provided correct answer.

Consider 10 i.i.d. random variables 𝑋 1 , 𝑋 2 , . . . , 𝑋 10 , each with expected value 0 and variance 5. What is the variance of the sample mean? That is, what is 𝑉 𝑎 𝑟 ( 𝑋 ¯ 10 ) ?单项选择题

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