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Chapter 8: Q 1E (page 484)

Suppose thatXhas thetdistribution withmdegrees of freedom(m >2). Show that Var(X)=m/(m鈭�2).

Hint:To evaluate\({\bf{E}}\left( {{{\bf{X}}^{\bf{2}}}} \right)\), restrict the integral to the positive half of the real line and change the variable fromxto

\({\bf{y = }}\frac{{\frac{{{{\bf{x}}^{\bf{2}}}}}{{\bf{m}}}}}{{{\bf{1 + }}\frac{{{{\bf{x}}^{\bf{2}}}}}{{\bf{m}}}}}\)

Compare the integral with the p.d.f. of a beta distribution. Alternatively, use Exercise 21 in Sec. 5.7.

Short Answer

Expert verified

\(Var\left( X \right) = \frac{m}{{m - 2}}\) . Proved

Step by step solution

01

Given information

A random variable X has the t- distribution with m degrees of freedom and m is greater than 2. So, the probability density function is,

\(f\left( x \right) = \frac{{\Gamma \left( {\frac{{m + 1}}{2}} \right)}}{{{{\left( {m\pi } \right)}^{\frac{1}{2}}}\Gamma \left( {\frac{m}{2}} \right)}}{\left( {1 + \frac{{{x^2}}}{m}} \right)^{ - \frac{{\left( {m + 1} \right)}}{2}}}\;for\; - \infty < x < \infty \) .

02

Evaluate the expectation of the random variable

As the variable X is from t-distribution, so,\(E\left( X \right) = 0\)

Let us consider the square of the expectation of the random variable\(E\left( {{X^2}} \right)\)

So,

\(\begin{align}E\left( {{X^2}} \right) &= c\int_{ - \infty }^\infty {{x^2}{{\left( {1 + \frac{{{x^2}}}{m}} \right)}^{\frac{{ - \left( {m + 1} \right)}}{2}}}dx} \\ &= 2c\int_{ - \infty }^\infty {{x^2}{{\left( {1 + \frac{{{x^2}}}{m}} \right)}^{\frac{{ - \left( {m + 1} \right)}}{2}}}dx} \end{align}\)

Where\(c = \frac{{\Gamma \left( {\frac{{m + 1}}{2}} \right)}}{{{{\left( {m\pi } \right)}^{\frac{1}{2}}}\Gamma \left( {\frac{m}{2}} \right)}}\)

Now, there is provided that,

\(\begin{align}y &= \frac{{\frac{{{x^2}}}{m}}}{{1 + \frac{{{x^2}}}{m}}}\\x &= {\left( {\frac{{my}}{{1 - y}}} \right)^{\frac{1}{2}}}\\\frac{{dx}}{{dy}} &= \frac{{\sqrt m }}{2}{y^{ - \frac{1}{2}}}{\left( {1 - y} \right)^{ - \frac{3}{2}}}\end{align}\)

Now by substituting the value of x and dx in \(E\left( {{X^2}} \right)\), we get,

\(\begin{align}E\left( {{X^2}} \right) &= 2c\int_{ - \infty }^\infty {{x^2}{{\left( {1 + \frac{{{x^2}}}{m}} \right)}^{\frac{{ - \left( {m + 1} \right)}}{2}}}dx} \\ &= \sqrt m c\int_0^1 {\frac{{my}}{{\left( {1 - y} \right)}}{{\left( {1 + \frac{y}{{1 - y}}} \right)}^{\frac{{ - \left( {m + 1} \right)}}{2}}}{y^{ - \frac{1}{2}}}{{\left( {1 - y} \right)}^{ - \frac{3}{2}}}dy} \\ &= {m^{\frac{3}{2}}}c\frac{{\Gamma \left( {\frac{3}{2}} \right)\Gamma \left( {\frac{{\left( {m - 2} \right)}}{2}} \right)}}{{\Gamma \left( {\frac{{\left( {m + 1} \right)}}{2}} \right)}}\\ &= m{\pi ^{ - \frac{1}{2}}}\Gamma \left( {\frac{3}{2}} \right)\frac{{\Gamma \left( {\frac{{\left( {m - 2} \right)}}{2}} \right)}}{{\Gamma \left( {\frac{m}{2}} \right)}}\\ &= m{\pi ^{ - \frac{1}{2}}}\left( {\frac{1}{2}\sqrt \pi } \right)\frac{1}{{\left( {\frac{{\left( {m - 2} \right)}}{2}} \right)}}\\ &= \frac{m}{{m - 2}}\end{align}\)

03

Calculate the variance

The variance of the variable is,

\(\begin{align}Var\left( X \right) &= E\left( {{X^2}} \right) - E\left( X \right)\\ &= \frac{m}{{m - 2}} - 0\\ &= \frac{m}{{m - 2}}\end{align}\)

Therefore, \(Var\left( X \right) = \frac{m}{{m - 2}}\) . Proved.

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Most popular questions from this chapter

Suppose that a random sampleX1, . . . , Xnis to be taken from the uniform distribution on the interval (0, 胃) and that胃is unknown. How large must a random sample be taken in order\({\bf{P}}\left( {{\bf{|max}}\left\{ {{{\bf{X}}_{\bf{1}}}{\bf{,}}...{\bf{,}}{{\bf{X}}_{\bf{n}}}} \right\}{\bf{ - \theta |}} \le {\bf{0}}{\bf{.10}}} \right) \ge {\bf{0}}{\bf{.95}}\) for all possible胃?

Question:Suppose that\({{\bf{X}}_{\bf{1}}}{\bf{,}}...{\bf{,}}{{\bf{X}}_{\bf{n}}}\)form a random sample from a distribution for which the p.d.f. or the p.f. is f (x|胃 ), where the value of the parameter 胃 is unknown. Let\({\bf{X = }}\left( {{{\bf{X}}_{\bf{1}}}{\bf{,}}...{\bf{,}}{{\bf{X}}_{\bf{n}}}} \right)\)and let T be a statistic. Assuming that 未(X) is an unbiased estimator of 胃, it does not depend on 胃. (If T is a sufficient statistic defined in Sec. 7.7, then this will be true for every estimator 未. The condition also holds in other examples.) Let\({{\bf{\delta }}_{\bf{0}}}\left( {\bf{T}} \right)\)denote the conditional mean of 未(X) given T.

a. Show that\({{\bf{\delta }}_{\bf{0}}}\left( {\bf{T}} \right)\)is also an unbiased estimator of 胃.

b. Show that\({\bf{Va}}{{\bf{r}}_{\bf{\theta }}}\left( {{{\bf{\delta }}_{\bf{0}}}} \right) \le {\bf{Va}}{{\bf{r}}_{\bf{\theta }}}\left( {\bf{\delta }} \right)\)for every possible value of 胃. Hint: Use the result of Exercise 11 in Sec. 4.7.

Suppose that \({{\bf{X}}_{\bf{1}}}{\bf{, \ldots ,}}{{\bf{X}}_{\bf{n}}}\)form a random sample from the normal distribution with unknown mean 渭and unknown variance \({{\bf{\sigma }}^{\bf{2}}}\), and let the random variableLdenote the length of the shortest confidence interval for渭that can be constructed from the observed values in the sample. Find the value of \({\bf{E}}\left( {{{\bf{L}}^{\bf{2}}}} \right)\)for the following values of the sample sizenand the confidence coefficient\(\gamma \):

\(\begin{align}{\bf{a}}{\bf{.n = 5,}}\gamma {\bf{ = 0}}{\bf{.95}}\\{\bf{b}}{\bf{.n = 10,}}\gamma {\bf{ = 0}}{\bf{.95}}\\{\bf{c}}{\bf{.n = 30,}}\gamma {\bf{ = 0}}{\bf{.95}}\\{\bf{d}}{\bf{.n = 8,}}\gamma {\bf{ = 0}}{\bf{.90}}\\{\bf{e}}{\bf{.n = 8,}}\gamma {\bf{ = 0}}{\bf{.95}}\\{\bf{f}}{\bf{.n = 8,}}\gamma {\bf{ = 0}}{\bf{.99}}\end{align}\)

Suppose that a point(X, Y, Z)is to be chosen at random in three-dimensional space, whereX,Y, andZare independent random variables, and each has the standard normal distribution. What is the probability that the distance from the origin to the point will be less than 1 unit?

Question:Suppose that a random variable X has a normal distribution for which the mean 渭 is unknown (鈭掆垶 <渭< 鈭�) and the variance 蟽2 is known. Let\({\bf{f}}\left( {{\bf{x}}\left| {\bf{\mu }} \right.} \right)\)denote the p.d.f. of X, and let\({\bf{f'}}\left( {{\bf{x}}\left| {\bf{\mu }} \right.} \right)\)and\({\bf{f''}}\left( {{\bf{x}}\left| {\bf{\mu }} \right.} \right)\)denote the first and second partial derivatives with respect to 渭. Show that

\(\int_{{\bf{ - }}\infty }^\infty {{\bf{f'}}\left( {{\bf{x}}\left| {\bf{\mu }} \right.} \right)} {\bf{dx = 0}}\,\,{\bf{and}}\,\,\int_{{\bf{ - }}\infty }^\infty {{\bf{f''}}\left( {{\bf{x}}\left| {\bf{\mu }} \right.} \right)} {\bf{dx = 0}}{\bf{.}}\).
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