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Chapter 3: Q12E (page 117)

Suppose that the c.d.f. of a random variableXis as

follows:

F(x)=forx鈮�3,

1 forx >3.

For the c.d.f. in Exercise 6, find the quantile function.

Short Answer

Expert verified

The first quantile is 1.96

The second quantile is 2.26

The third quantile is 2.44

Step by step solution

01

Given the information

Given cdf given from continuous distribution. The cdf is given for different ranges.

02

State the quantile

Quantile function we defined as

F(x) = p

X = F-1(p)

03

Compute the quantile function

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Most popular questions from this chapter

Let W denote the range of a random sample of nobservations from the uniform distribution on the interval[0, 1]. Determine the value of

\({\bf{Pr}}\left( {{\bf{W > 0}}{\bf{.9}}} \right)\).

Suppose that the p.d.f. of a random variableXis as follows:

\(f\left( x \right) = \left\{ \begin{array}{l}\frac{1}{8}x\;\;for\;0 \le x \le 4\\0\;\;\;\;otherwise\end{array} \right.\)

a. Find the value oftsuch that Pr(X鈮�t)=1/4.

b. Find the value oftsuch that Pr(X鈮�t)=1/2.

Question:Suppose thatXandYhave a continuous joint distribution

for which the joint p.d.f. is defined as follows:

\({\bf{f}}\left( {{\bf{x,y}}} \right){\bf{ = }}\left\{ \begin{array}{l}\frac{{\bf{3}}}{{\bf{2}}}{{\bf{y}}^{\bf{2}}}\;{\bf{for}}\;{\bf{0}} \le {\bf{x}} \le {\bf{2}}\;{\bf{and}}\;{\bf{0}} \le {\bf{y}} \le {\bf{1}}\\{\bf{0}}\;\,{\bf{otherwise}}\end{array} \right.\)

a. Determine the marginal p.d.f.鈥檚 ofXandY.

b. AreXandYindependent?

c. Are the event{X<1}and the event\(\left\{ {{\bf{Y}} \ge \frac{{\bf{1}}}{{\bf{2}}}} \right\}\)independent?

Consider the situation described in Example 3.7.14. Suppose that \(\) \({X_1} = 5\) and\({X_2} = 7\)are observed.

a. Compute the conditional p.d.f. of \({X_3}\) given \(\left( {{X_1},{X_2}} \right) = \left( {5,7} \right)\).

b. Find the conditional probability that \({X_3} > 3\)given \(\left( {{X_1},{X_2}} \right) = \left( {5,7} \right)\)and compare it to the value of \(P\left( {{X_3} > 3} \right)\)found in Example 3.7.9. Can you suggest a reason why the conditional probability should be higher than the marginal probability?

For the conditions of Exercise 9, determine the value of \({\bf{Pr}}\left( {{{\bf{Y}}_{\bf{1}}} \le {\bf{0}}{\bf{.1}}\;{\bf{and}}\;{\bf{Y}}_{\bf{n}}^{} \ge {\bf{0}}{\bf{.8}}} \right)\).
See all solutions

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