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Chapter 4: Q12E (page 234)

Let X have the binomial distribution with parameters5 and 0.3. Find the IQR of X. Hint: Return to Example3.3.9 and Table 3.1.

Short Answer

Expert verified

The IQR of X is 1.

Step by step solution

01

Given information

X follows the binomial distribution with parameters 5 and 0.3.

02

Find the inter-quartile range of X

\(X \sim Binomial\left( {5,0.3} \right)\).

So,\(F\left( x \right) = \sum\limits_{x = 0}^x {\left( {\begin{aligned}{*{20}{c}}5\\x\end{aligned}} \right){{\left( {0.3} \right)}^x}{{\left( {0.7} \right)}^{5 - x}}} ;0 \le x \le 5\).

Now, let\(F\left( {{x_1}} \right) = 0.25\)and\(F\left( {{x_3}} \right) = 0.75\).

From example 3.3.9 and Table 3.1, we have observed that\({x_1} = 1\)and\({x_3} = 2\).

The inter-quartile range of X is derived as shown below.

\(\begin{aligned}{c}{F^{ - 1}}\left( {0.75} \right) - {F^{ - 1}}\left( {0.25} \right) &= {x_3} - {x_1}\\ &= 2 - 1\\ &= 1\end{aligned}\)

Therefore, the IQR of X is 1.

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Most popular questions from this chapter

Suppose that X and Y have a continuous joint distribution for which the joint p.d.f. is as follows:

\({\bf{f}}\left( {{\bf{x,y}}} \right){\bf{ = }}\left\{ {\begin{align}{}{{\bf{x + y}}}&{{\bf{for}}\,\,{\bf{0}} \le {\bf{x}} \le {\bf{1}}\,\,{\bf{and}}\,\,{\bf{0}} \le {\bf{y}} \le {\bf{1,}}}\\{\bf{0}}&{{\bf{otherwise}}{\bf{.}}}\end{align}} \right.\).

Find\({\bf{E}}\left( {{\bf{Y}}\left| {\bf{X}} \right.} \right)\)and\({\bf{Var}}\left( {{\bf{Y}}\left| {\bf{X}} \right.} \right)\).

Suppose that a person's score X on a mathematics aptitude test is a number in the interval\(\left( {0,1} \right)\)and that his score Y on a music aptitude test is also a number in the interval\(\left( {0,1} \right)\)Suppose also that in the population of all college students in the United States, the scores X and Y are distributed in accordance with the following joint p.d.f:

\(f\left( {x,y} \right) = \left\{ \begin{align}\frac{2}{5}\left( {2x + 3y} \right)\;\;\;\;\;\;\;for\,0 \le x \le 1\,and0 \le x \le 1\\0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;otherwise\end{align} \right.\)

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b. What predicted value of his score on the mathematics test has the smallest M.A.E.?

Prove the following extension of Theorem 4.4.1: If \(E\left( {{{\left| X \right|}^a}} \right) < \infty \) for some positive number a, then \(E\left( {{{\left| X \right|}^b}} \right) < \infty \) for every positive number \(b < a\). Give the proof for the case in which X has a discrete distribution.

Consider a coin for which the probability of obtaining a head on each given toss is 0.3. Suppose that the coin is to be tossed 15 times, and let X denote the number of heads that will be obtained.

  1. What prediction of X has the smallest M.S.E?
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Let\({\bf{\alpha > 0}}\). A decision-maker has a utility function for money of the form

\({\bf{U}}\left( {\bf{x}} \right){\bf{ = }}\left\{ {\begin{align}{}{{{\bf{x}}^{\bf{\alpha }}}}&{{\bf{if}}\,{\bf{x > 0,}}}\\{\bf{x}}&{{\bf{if}}\,{\bf{x}} \le {\bf{0}}{\bf{.}}}\end{align}} \right.\)

Suppose that this decision maker is trying to decide whether or not to buy a lottery ticket for \(1. The lottery ticket pays \)500 with a probability of 0.001, and it pays $0 with a probability of 0.999. What would the values of 伪 have to be for this decision-maker to prefer buying the ticket to not buying it?
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