Week 20: Chi-Square Distribution: Find Expected Value of the Error

In the lecture “Find Expected Value of the Error”, I feel S(n-1)^2 value shown is incorrect.

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The summation should be (i=1 to n-1) instead of (i=1 to n) so,
S(n-1)^2 not equal to (n / (n-1))*Sn^2.

Please let me know if my conjecture is correct or wrong with explanation.

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Hi @Abhineet,
No, the formula is correct.
Note that the earlier estimate (S_n)² was doing an underestimate, considering which we decided to take the denominator as (n-1). If we change the summation limit as well, it would again start underestimating.

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Here S_(n-1) ^2 term is said to be a Chi square distribution with (n-1) degrees of freedom.
But the summation in S_(n-1) ^2 is for ‘n’ samples so the degrees of freedom should be ‘n’ right?

Lecture Name: Recap and Statistics of S2

Hi @Abhineet,
The n-degrees of freedom is itself decomposed into n-1 and 1 degree of freedom terms.

I understand that the ‘n’ degrees of freedom from LHS are decomposed into ‘n-1’ and ‘1’ degrees of freedom terms from RHS.
But how S_(n-1) ^2 represents ‘n-1’ degrees of freedom? This is causing some confusion for me.
As I said before, the summation in S_(n-1) ^2 is for ‘n’ samples so it has ‘n’ terms of (Z_i)^2. So, does it not make degrees of freedom for S_(n-1) ^2 as ‘n’?

Yes, but the main difference between S_n-1 and S_n is the denominator term…

Sir how does the denominator being (n-1) make the degrees of freedom as (n-1) ? Degrees of freedom depends on the number of Z score terms right, and the number of Z score terms are ‘n’ in S_(n-1) ^2.
I just want to know why S_(n-1) ^2 is said to have (n-1) degrees of freedom. When Pratyush sir introduced Degrees of freedom concept, he had said that degrees of freedom depends on the number of Z score terms or indirectly the number of (X_i)^2 terms. Equation shown was, Q = Z1^2 + Z2^2 + Z3^2 + … + Zn^2.
Comparing the equation of ‘Q’ with S_(n-1) ^2, proves that S_(n-1) ^2 has ‘n’ degrees of freedom.

Look, I get you… but recall that the degrees of freedom has a link with the number of independent random variables that we can take.

And If we get back to the explanations, in the lectures it’s also explained that how there are only n-1 independent random variables possible.
You must checkout that derivation to get better of an understanding regarding this.

Ok, thanks…doubt cleared👍

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