The law of large numbers describes the result of performing the same experiment a large number of times. Let be examples i.i.d. drawn from a distribution . Let be a function. Then which equation reflects the law of large numbers.

A.
lim
𝑛
→
∞
∑
𝑖
=
1
𝑛
𝑓
(
𝑥
𝑖
)
=
𝐸
𝑋
∼
𝐷
[
𝑓
(
𝑋
)
]
.

B.
lim
𝑛
→
∞
1
𝑛
∑
𝑖
=
1
𝑛
𝑓
(
𝑥
𝑖
)
=
𝐸
𝑋
∼
𝐷
[
𝑓
(
𝑋
)
]
.

C.
lim
𝑛
→
∞
∑
𝑖
=
1
𝑛
𝑓
(
𝑥
𝑖
)
=
𝑓
(
𝐸
𝑋
∼
𝐷
[
𝑋
]
)
.

D.
lim
𝑛
→
∞
1
𝑛
∑
𝑖
=
1
𝑛
𝑓
(
𝑥
𝑖
)
=
𝑓
(
𝐸
𝑋
∼
𝐷
[
𝑋
]
)
.

E. None of the above equations is true.

Question

The law of large numbers describes the result of performing the same experiment a large number of times. Let  be  examples i.i.d. drawn from a distribution . Let  be a function. Then which equation reflects the law of large numbers.

A. 
lim
𝑛
→
∞
∑
𝑖
=
1
𝑛
𝑓
(
𝑥
𝑖
)
=
𝐸
𝑋
∼
𝐷
[
𝑓
(
𝑋
)
]
.

B. 
lim
𝑛
→
∞
1
𝑛
∑
𝑖
=
1
𝑛
𝑓
(
𝑥
𝑖
)
=
𝐸
𝑋
∼
𝐷
[
𝑓
(
𝑋
)
]
.

C. 
lim
𝑛
→
∞
∑
𝑖
=
1
𝑛
𝑓
(
𝑥
𝑖
)
=
𝑓
(
𝐸
𝑋
∼
𝐷
[
𝑋
]
)
.

D. 
lim
𝑛
→
∞
1
𝑛
∑
𝑖
=
1
𝑛
𝑓
(
𝑥
𝑖
)
=
𝑓
(
𝐸
𝑋
∼
𝐷
[
𝑋
]
)
.

E. None of the above equations is true.

BlackTom AI · Accepted Answer

The task presents a question about the law of large numbers (LLN) and provides a set of answer choices A–E, but the answer_options field is empty, so we don’t have the explicit text of each option to analyze individually. Nevertheless, we can reason about the LLN in general and evaluate the likely structure of the statements.

First, recall the standard statement of the law of large numbers in this context: if X1, X2, ..., Xn are i.i.d. samples from a distribution D, and f is a function with a finite expectation E[f(X)], then the sample average converges to the expected value as n → ∞:
lim_{n→∞} (1/n) ∑_{i=1}^n f(X_i) = E[f(X)].

With that in mind, consider what a correct option would look like:
- It should involve the limit of the average (1/n) ∑ f(X_i).
- It should equate this limit to the expectation E[f(X)], not E[X] or some other quantity, unless properly transformed (e.g., if the option explicitly states E[f(X)], not something like E[X] or a distributional symbol that misrepresents the quantity).

Now, given the common forms of LLN statements, evaluate the typical options (in a general sense):
- If an option states lim_{n→∞} (1/n) ∑ f(X_i) = E[f(X)], that aligns with the LLN, since the left-hand side is the average of the function values and the right-hand side is the expected value of the function applied to a new draw from D. This would be correct.
- If an option omits the (1/n) factor and writes lim_{n→∞} ∑ f(X_i) (without normalization), that is generally incorrect, because the sum diverges or does not converge to a meaningful constant in the LLN context.
- If an option writes E[X] or E[X] ∼ D [ f(X) ] in a way that mislabels the quantity (for example, using E[X] instead of E[f(X)]), that would be incorrect, since the LLN concerns E[f(X)], not E[X] unless f is the identity function.
- If an option asserts a limit that does not exist or misattributes the limit to a different distributional quantity, that would be incorrect as well.

Based on the standard LLN, the most faithful statement would involve the normalized sum (1/n) ∑ f(X_i) converging to E[f(X)]. Among typical options, that’s usually the correct form or the closest to it. The other options are incorrect for reasons such as missing normalization, misidentifying the target expectation, or miswriting the limiting quantity.

Because the actual option texts are missing in this input, we cannot single out which letter corresponds exactly to the LLN in this specific question. However, given the conventional LLN form, the option that presents lim_{n→∞} (1/n) ∑ f(X_i) = E[f(X)] is the correct one, and any option omitting the 1/n factor or substituting E[X] for E[f(X)] would be incorrect.

If you can provide the explicit texts for options A–E, I can give a precise, option-by-option explanation for why each choice is correct or incorrect, with detailed reasoning for the LLN in this context.

COMP4328 COMP5328 COMP8328 (NE) Week 4 quiz

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