As above, let Y1,…,Yn denote the i 'th number in the binary data set.
Recall that Y1,…,Yn are assumed to be independent and identically distributed (i.i.d.) as some distribution Y . In the future, we will abbreviate this assumption with the notation Y1,…,Yn∼iidY .
Which of the following converges to E[Yi]=E[Y] as n→∞ ?
(Choose all that apply.)
a) total number of 1'sn
b) Yn
c) Median(Y1,…,Yn)
d) 1n∑i=1nYi
a) total number of 1'sn
b) Yn
Neither c) Median(Y1,…,Yn) nor d) 1n∑i=1nYi converge to E[Yi]=E[Y] as n→∞. However, it is assumed that Y1,…,Yn converge to E[Yi]=E[Y] as n→∞.
To determine which of the given options converges to E[Yi]=E[Y] as n approaches infinity, we can analyze each option individually:
a) total number of 1'sn:
The total number of 1's, n, in the binary dataset does not converge to E[Yi]. It represents the count of occurrences and provides no information about the expected value of Y.
b) Yn:
Yn represents the i'th number in the dataset and does not converge to E[Yi]. It refers to a specific value from the dataset rather than the expected value.
c) Median(Y1,…,Yn):
The median of Y1,…,Yn does not necessarily converge to E[Yi]. The median represents the middle value in a sorted dataset, and its convergence to the expected value depends on the shape of the distribution.
d) 1n∑i=1nYi:
The expression 1n∑i=1nYi represents the sample mean of Y1,…,Yn. As n approaches infinity, the sample mean converges in probability to the expected value E[Yi] or E[Y]. Therefore, this option converges to E[Yi].
Based on the analysis, the option that converges to E[Yi] as n approaches infinity is d) 1n∑i=1nYi.
In order to determine which of the given options converges to the expectation E[Y] as n approaches infinity, let's analyze each option:
a) The total number of 1's in the data set, denoted by n. This does not converge to E[Y] because it does not involve any calculations or operations on the data itself. It only represents the total count of 1's, which doesn't provide information about the expectation.
b) Yn, which represents the i-th number in the data set. By definition, Yn is a single value from the data set, and it does not involve any summation or averaging of observations. Therefore, Yn does not converge to E[Y].
c) The median of Y1,...,Yn. The median is the middle value when the data set is sorted in ascending order. The median is not guaranteed to converge to E[Y] as n approaches infinity because it only captures a single value from the data set, not the overall expectation.
d) 1/n * sum(i=1 to n) Yi. This option involves taking the average of all the data points, denoted by Yi, as n approaches infinity. This is the sample average of the data set, and according to the Law of Large Numbers, it converges to the population expectation E[Y] as n approaches infinity. Therefore, this option, 1/n * sum(i=1 to n) Yi, converges to E[Y] as n approaches infinity.
In summary, the option that converges to E[Y] as n approaches infinity is d) 1/n * sum(i=1 to n) Yi.