Question

Consider a random variable X following the exponential distribution X ~ f(x), where f(x) = ae^( -ax) for x > 0 and 0 otherwise, a > 0. Derive its moment-generating function MX(t) and specify its domain (where it is defined or for what t does the integral exist). Use it to compute the first four non-central moments of X and then derive the general formula for the nth non-central moment for any positive integer n. Also, write down the expression for variance, skewness, and kurtosis.

Answer #1

The pdf of X is

The MGF will be

So MGF is

-------------------

Differentiating MGF with respect to t gives:

Putting t=0 gives

-------------

Differentiating MGF again with respect to t gives:

Putting t=0 gives

Differentiating MGF again with respect to t gives:

Putting t=0 gives

Differentiating MGF again with respect to t gives:

Putting t=0 gives

The general expression is

The variance will be

The skewness is

The Kurtosis is

Given the exponential distribution f(x) = λe^(−λx), where λ >
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M(t). Why is the
series expansion relevant in terms of calculating moments?

Define the nth moment of the random variable X. Define the nth
central moment of a random variable X. Finally, define the moment
generating function, M(t). Write down a few terms of the series
expansion of a general M(t). Why is the series expansion relevant
in terms of calculating moments?

Define the nth moment of the random variable X. Define the nth
central moment of a random variable X. Finally, define the moment
generating function, M(t). Write down a few terms of the series
expansion of a general M(t). Why is the series expansion relevant
in terms of calculating moments

Let X have exponential density f(x) = λe−λx if x >
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If X is a discrete random variable with uniform distribution,
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1. Please find the probability distribution of (X + Y)/2 and
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2. Please find variance of (X + Y)/2

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b. use chebychev's inequality to find a bound on this
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distribution with rate λ = 7 Thus the pdf of X is f(x) = λ
e−λx for 0 ≤ x where λ = 7.
a) Using the f(x) above and the R integrate function calculate the
expected value of X.
b) Using the f(x) above and the R integrate function calculate the
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c) Using the dexp function and the R integrate command calculate
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(b) (3pts) Find the cumulative distribution function (cdf) F(x)
of X ⇠ U(0, 2). (Caution: Please specify the function values for
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