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Chapter 4: Q16E (page 248)

Let X be a random variable. Suppose that there exists a number m such that \(\Pr \left( {X < m} \right) = \Pr \left( {X > m} \right)\).Prove that m is a median of the distribution ofX.

Short Answer

Expert verified

m is a median of X.

Step by step solution

01

Given information

X is a random variable. There exists a number m such that \(\Pr \left( {X < m} \right) = \Pr \left( {X > m} \right)\).

02

Prove thatm is a median of X

We know that\(\Pr \left( {X < m} \right) = \Pr \left( {X > m} \right).\)

\(\begin{array}{l}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,Now,\,\,\Pr \left( {X < m} \right) = \Pr \left( {X > m} \right)\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\Pr \left( {X < m} \right) = 1 - \Pr \left( {X \le m} \right)\\\Pr \left( {X < m} \right) + \Pr \left( {X \le m} \right) = 1\\\Pr \left( {X \le m} \right) + \Pr \left( {X \le m} \right) \ge 1\;\left( \because {\Pr \left( {X < m} \right) \le \Pr \left( {X \le m} \right)} \right)\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2\Pr \left( {X \le m} \right) \ge 1\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\Pr \left( {X \le m} \right) \ge \frac{1}{2}.................\left( 1 \right)\end{array}\)

\(\begin{array}{l}\,\,\,\,\,\,\,\,\,\,\,\,Again,\,\,\Pr \left( {X > m} \right) = \Pr \left( {X < m} \right)\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\Pr \left( {X > m} \right) = 1 - \Pr \left( {X \ge m} \right)\\\Pr \left( {X > m} \right) + \Pr \left( {X \ge m} \right) = 1\\\Pr \left( {X \ge m} \right) + \Pr \left( {X \ge m} \right) \ge 1\;\left( \because {\Pr \left( {X \ge m} \right) \ge \Pr \left( {X > m} \right)} \right)\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2\Pr \left( {X \ge m} \right) \ge 1\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\Pr \left( {X \ge m} \right) \ge \frac{1}{2}.........................\left( 2 \right)\end{array}\)

From equations (1) and (2), it is clear that m satisfies both the conditions of being a median.

Hence, m is a median of X.

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Most popular questions from this chapter

Suppose that\({\bf{X}}\)and\({\bf{Y}}\)are random variables such that

\({\bf{Var}}\left( {\bf{X}} \right){\bf{ = 9}}\),\({\bf{Var}}\left( {\bf{Y}} \right){\bf{ = 4}}\),and\({\bf{\rho }}\left( {{\bf{X,Y}}} \right){\bf{ = - }}\frac{{\bf{1}}}{{\bf{6}}}\).Determine

(a)\({\bf{Var}}\left( {{\bf{X + Y}}} \right)\)and(b)\({\bf{Var}}\left( {{\bf{X - 3Y + 4}}} \right)\).

Suppose thatXandYhave a continuous joint distributionfor which the joint p.d.f. is as follows:

\({\bf{f}}\left( {{\bf{x,y}}} \right){\bf{ = }}\left\{ \begin{array}{l}{\bf{12}}{{\bf{y}}^{\bf{2}}}\;{\bf{for}}\;{\bf{0}} \le {\bf{y}} \le {\bf{x}} \le {\bf{1}}\\{\bf{0}}\;{\bf{otherwise}}\end{array} \right.\)

Find the value ofE(XY).

Suppose that the distribution of a random variable Xis symmetric with respect to the point \(x = 0\) and that \(E\left( {{X^4}} \right) < \infty \).Show that \(E\left( {{{\left( {X - d} \right)}^4}} \right)\)is minimized by the value \(d = 0\).

Suppose that \({\bf{0 < Var}}\left( {\bf{X}} \right){\bf{ < }}\infty \)and \({\bf{0 < Var}}\left( Y \right){\bf{ < }}\infty \) Show that if \({\bf{E}}\left( {{\bf{X|Y}}} \right)\)is constant for all values ofY, thenXandYare uncorrelated.

LetXbe a random variable for which \(E\left( X \right) = \mu \), and\(Var\left( X \right) = {\sigma ^2}\), and let c be an arbitrary constant. Show that \(E\left( {{{\left( {X - c} \right)}^2}} \right) = {\left( {\mu - c} \right)^2} + {\sigma ^2}\).
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