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Chapter 4: Q 1E (page 255)

Suppose that the pair(X, Y )is uniformly distributed on the interior of a circle of radius 1. Compute蚁(齿,驰).

Short Answer

Expert verified

\(\rho \left( {X,Y} \right) = 0\)

Step by step solution

01

Given information

\(\left( {X,Y} \right)\)is uniformly distributed interior of a circle of radius 1.

02

Calculate \(\rho \left( {X,Y} \right)\)

The location of the circle makes no difference since it only affects the means of \(X\)and\(Y\).

Let, the circle is centred at\(\left( {0,0} \right)\)

\(\begin{align}Cov\left( {X,Y} \right) &= 0\\ \Rightarrow \rho \left( {X,Y} \right) &= 0\end{align}\)

Hence,\(\rho \left( {X,Y} \right) = 0\).

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Most popular questions from this chapter

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:

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

a. If a college student is selected randomly, what predicted value of his score on the music test has the smallest M.S.E.?

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

Suppose that one letter is to be selected at random from the 30 letters in the sentence given in Exercise 4. If Y denotes the number of letters in the word in which the selected letter appears, what is the value of E (Y)?

Suppose that a random variable X has a continuous distribution for which the pdf is as follows:

\(f\left( x \right) = \left\{ {\begin{aligned}{{}{}}{{e^{ - x}}}&{{\rm{for }}x > 0}\\0&{{\rm{otherwise}}}\end{aligned}} \right.\)

Determine all the medians of this distribution.

Suppose that three random variables\({{\bf{X}}_{\bf{1}}}{\bf{,}}{{\bf{X}}_{\bf{2}}}{\bf{,}}{{\bf{X}}_{\bf{3}}}\)froma random sample from a distribution for which the meanis 5. Determine the value of

\({\bf{E}}\left( {{\bf{2}}{{\bf{X}}_{\bf{1}}}{\bf{ - 3}}{{\bf{X}}_{\bf{2}}}{\bf{ + }}{{\bf{X}}_{\bf{3}}}{\bf{ - 4}}} \right)\).

Consider again the conditions of Exercise 2, but suppose now that X has a discrete distribution with c.d.f.\(F\left( x \right)\)F (x),rather than a continuous distribution. Show that the conclusion of Exercise 2 still holds
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