should be the result of dividing by n - 1 instead of n. So, recall that if we have n samples X1,..., Xn, the (x Here, XX, is the sample mean. Suppose all the samples {X}, are independent and mean, and let M2=E[X] denote their second moment
should be the result of dividing by n - 1 instead of n. So, recall that if we have n samples X1,..., Xn, the (x Here, XX, is the sample mean. Suppose all the samples {X}, are independent and mean, and let M2=E[X] denote their second moment
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![Problem 3. In this problem, we're actually going to do the arithmetic to show why the sample variance
should be the result of dividing by n - 1 instead of n.
So, recall that if we have n samples X1,..., Xn, the sample variance is defined by:
72
n-1
Σ - Χ
Xx
i=1
Here, X=1X; is the sample mean.
Suppose all the samples {X}; are independent and identically distributed. Let =E[X] denote their
mean, and let M₂ =E[X] denote their second moment. Thus, the variance of X; is M2 -μ².
(a) For Part (a), you are asked to calculate the expected value of this quantity.
71
Σ
(Xi
Plug in the equation for the sample mean to the above expression, and calculate the expected value.
Hint: Here's a few nice things to recall that will be useful for this problem. First, expectation is
linear, i.e.:
ΕΣΖ = ΣΕ[Ζι]
-
Li=1
i=1
Second, independence implies that E[X;X;] = E[X;]E[X;] if i #j.
As another hint, note that:
-(Eα)-E (~(2~))
(a1a2+...an) Σ a}/2/3 + ]
=
i=1
(b) Next, suppose we actually know the mean =E[X], and we are allowed to use it directly estimate
the variance. Then we would want to use the quantity:
72
Σ(Χ. - μ)
i=1
Calculate the expected value of the above expression.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F423822cb-5d3f-48f4-83a4-f0ba428ae79c%2F9abe3a1c-58e1-439d-a464-9f8128c5f3a1%2F5z4dom8_processed.png&w=3840&q=75)
Transcribed Image Text:Problem 3. In this problem, we're actually going to do the arithmetic to show why the sample variance
should be the result of dividing by n - 1 instead of n.
So, recall that if we have n samples X1,..., Xn, the sample variance is defined by:
72
n-1
Σ - Χ
Xx
i=1
Here, X=1X; is the sample mean.
Suppose all the samples {X}; are independent and identically distributed. Let =E[X] denote their
mean, and let M₂ =E[X] denote their second moment. Thus, the variance of X; is M2 -μ².
(a) For Part (a), you are asked to calculate the expected value of this quantity.
71
Σ
(Xi
Plug in the equation for the sample mean to the above expression, and calculate the expected value.
Hint: Here's a few nice things to recall that will be useful for this problem. First, expectation is
linear, i.e.:
ΕΣΖ = ΣΕ[Ζι]
-
Li=1
i=1
Second, independence implies that E[X;X;] = E[X;]E[X;] if i #j.
As another hint, note that:
-(Eα)-E (~(2~))
(a1a2+...an) Σ a}/2/3 + ]
=
i=1
(b) Next, suppose we actually know the mean =E[X], and we are allowed to use it directly estimate
the variance. Then we would want to use the quantity:
72
Σ(Χ. - μ)
i=1
Calculate the expected value of the above expression.
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