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Chapter 1: Problem 26

Let \(X\) have the exponential pdf, \(f(x)=\beta^{-1} \exp \\{-x / \beta\\}, 0

Short Answer

Expert verified
The moment generating function (mgf) of \(X\) is \((1-\beta t)^{-1}\) for \(t<1/\beta\), the mean of \(X\) is \(\beta\), and the variance of \(X\) is 0.

Step by step solution

01

Find the Moment Generating Function (mgf)

The moment generating function (mgf) of a random variable X is defined by \(M(t)=E[e^{tX}]\) where E denotes expected value. So, for this exercise we will integrate the product of the exponential pdf and \(e^{tx}\) from 0 to infinity, namely: \n\n\[M(t)=\int_0^\infty \beta^{-1} \exp(-x/\beta)e^{tx} dx\]\n\nNote that this integral converges for \(t < 1/\beta\). After solving the integral, the mgf yields: \n\n\[M(t)=(1-\beta t)^{-1},\quad for\: t<1/\beta\]
02

Find the Mean and Variance of \(X\)

The mean (expected value) and variance of a random variable can be found from its mgf. The mean, denoted \(\mu\), is \(M'(0)\), where \(M'\) denotes the derivative of \(M\): \n\n\[\mu=M'(0)=\beta\]\n\nThe second moment of \(X\) is \(M''(0)\). Thus, \n\n\[M''(0)= \beta^2\]\n\nThe variance of \(X\), denoted \(Var(X)\) or \(\sigma^2\), is calculated as follows: \n\n\[Var(X) = M''(0) - [M'(0)]^2 = \beta^2-\beta^2 = 0\]

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Most popular questions from this chapter

$$ \text { Let } X \text { have the pmf } p(x)=1 / 3, x=-1,0,1 \text { . Find the pmf of } Y=X^{2} \text { . } $$

A mode of the distribution of a random variable \(X\) is a value of \(x\) that maximizes the pdf or pmf. If there is only one such \(x\), it is called the mode of the distribution. Find the mode of each of the following distributions: (a) \(p(x)=\left(\frac{1}{2}\right)^{x}, x=1,2,3, \ldots\), zero elsewhere. (b) \(f(x)=12 x^{2}(1-x), 0

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$$ \begin{aligned} &\text {Let } X \text { have the pmf } p(x)=\left(\frac{1}{2}\right)^{x}, x=1,2,3, \ldots, \text { zero elsewhere. Find the }\\\ &\mathrm{pmf} \text { of } Y=X^{3} \end{aligned} $$

$$ \begin{aligned} &\text {Let } X \text { have a pmf } p(x)=\frac{1}{3}, x=1,2,3, \text { zero elsewhere. Find the pmf of }\\\ &Y=2 X+1 \end{aligned} $$
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