1. Let X = E-1 Xi. Use the properties of the expectation and the variance to show that E(X)= µ and Var(X) = iid 2. Suppose X1,..., X, N(u,o²). Then the mgf of X; is given as 1 Мx, (t) 3 еxp (ut + i = 1,..., n. 1 Use the independence of the random variables X;, i = 1,..., n, to show that the mgf of X is Mx(t) = exp (ut + 2 n From the mgf of X, identify the distribution of X.
1. Let X = E-1 Xi. Use the properties of the expectation and the variance to show that E(X)= µ and Var(X) = iid 2. Suppose X1,..., X, N(u,o²). Then the mgf of X; is given as 1 Мx, (t) 3 еxp (ut + i = 1,..., n. 1 Use the independence of the random variables X;, i = 1,..., n, to show that the mgf of X is Mx(t) = exp (ut + 2 n From the mgf of X, identify the distribution of X.
A First Course in Probability (10th Edition)
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ISBN:9780134753119
Author:Sheldon Ross
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Transcribed Image Text:Q3. Expected value, variance and the mgf of the sample mean
Let X1,..., X, be independent and identically distributed random variables with mean E(X;) = µ and the
variance Var(X;) = o², i = 1,..., n.
1. Let X = E-1 X;. Use the properties of the expectation and the variance to show that
i%3D1
E(X) = µ
and
o2
Var(X)
=
iid
2. Suppose X1,..., X, N(4, o²). Then the mgf of X; is given as
1
Mx, (t) = exp (ut +o?t2
2
i = 1,..., n.
1
Use the independence of the random variables X;, i = 1,...,n, to show that the mgf of X is
Mx(t) = exp ( ut +
2 n
From the mgf of X, identify the distribution of X.
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