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Second Moment of Uniform Distribution – (With Proof)

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The second moment of the uniform distribution is given by the formula,

E(X2)= (b2+ba+a2)/3.

Let X be a random variable following uniform distribution with values lying between ‘a’ and ‘b’. The pdf of X is given by the formula,

f(x) = 1/(b-a) where, axb.

Proof:

E(X2) = abx2f(x)dx = abx2 * 1/(b-a) dx.

E(X2) = 1/(b-a) abx2 dx = 1/(b-a) * [(b3 -a3)/3].

E(X2) = 1/(b-a) * (b -a)(b2+ba+a2)/3 = (b2+ba+a2)/3.

To summarize we have,

Proof of second moment of uniform distribution

Proof (Using MGF) :

In order to find the second random moment of the uniform distribution using the MGF we should differentiate the moment generating function twice and then set t=0.

The MGF (Moment Generating Function) of the uniform distribution is given as,

Second Moment of Uniform Distribution using MGF step 1

On differentiating it twice we get,

Second Moment of Uniform Distribution using MGF step 2

Putting t=0 on both sides we get that,

Second Moment of Uniform Distribution using MGF step 3
Summary
Second Moment of Uniform Distribution - (With Proof)
Article Name
Second Moment of Uniform Distribution - (With Proof)
Description
The second moment of the uniform distribution is given by the formula, E(X2)= (b2+ba+a2)/3.
Publisher Name
allthingsstatistics.com

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