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Asked Dec 10, 2019
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Find the mean and variance of a random variable with moment generating function
M2(t) = exp [4(e' – 1)]
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Find the mean and variance of a random variable with moment generating function M2(t) = exp [4(e' – 1)]

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Expert Answer

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Step 1

Given that

The moment generating function of a random variable is

To find mean and variance:

Step 2

Mean:

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E(x') = M, (t)(0) M, (1)(0) = M, (?) .- dt M, (1) = exp[4(e' –1)] -M, dt (4(e' -1)) (e*). du (e*) =e" du (4(e -1)) = 4e' u = 4(e' -1) =e-).4e apply exp onent rule =e 4, (t) = 4e%o"-t}er dt

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Step 3

Now sub t=0 in differential equati...

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M. (t) = 4e%*-1}er dt subt =0 = 4e lo°-1}=0 e = 1 = de1-1) = 4el0) = 4e° E, (t) = 4

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