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Chapter 3: Q11E (page 117)

(x)=0 for x鈮�0,

1/9x2 for 0 < x鈮�3,

1 for x >3.

For the c.d.f. in Exercise 5, find the quantile function.

Short Answer

Expert verified

The first quantile is 1.18

The second quantile is 1.37

The third quantile is 1.50

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

From the given cdf for the range 0 < x 鈮� 3, the quantile function defined as

X2/9 = F(x) = p 鈥�.(1)

In the equation, (1) both sides logarithm and calculate to get

\begin{aligned}\log p=2\log x-\log 9\\\log x=\frac{1}{2}\log\left({9p}\right)\\x={e^{\frac{1}{2}\log\left({9p}\right)}}\end{aligned}

Therefore, we calculate the first quantile, second quantile, and third quantile.

Then for the first quantile p = 1/4 =25% in equation (2) to get

For the second quantile p = 1/2 = 50% in the equation, (2) to get

For the third quantile p= 3/4 =75% in the equation, (2) to get


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Most popular questions from this chapter

Suppose that\({X_1}...{X_n}\)are independent. Let\(k < n\)and let\({i_1}.....{i_k}\)be distinct integers between 1 and n. Prove that \(X{i_1}.....X{i_k}\)they are independent.

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?

Suppose that the p.d.f. of a random variable X is as

follows:\(f\left( x \right) = \left\{ \begin{array}{l}\frac{1}{2}x\,\,\,\,\,\,\,\,for\,0 < x < 2\\0\,\,\,\,\,\,\,\,\,\,\,\,otherwise\end{array} \right.\)

Also, suppose that \(Y = X\left( {2 - X} \right)\) Determine the cdf and the pdf of Y .

In a certain city, three newspapersA,B, andC,are published. Suppose that 60 percent of the families in the city subscribe to newspaperA, 40 percent of the families subscribe to newspaperB, and 30 percent subscribe to newspaperC. Suppose also that 20 percent of the families subscribe to bothAandB, 10 percent subscribe to bothAandC, 20 percent subscribe to bothBandC, and 5 percent subscribe to all three newspapersA,B, andC. Consider the conditions of Exercise 2 of Sec. 1.10 again. If a family selected at random from the city subscribes to exactly one of the three newspapers,A,B, andC, what is the probability that it isA?

In Example 3.8.4, the p.d.f. of \({\bf{Y = }}{{\bf{X}}^{\bf{2}}}\) is much larger for values of y near 0 than for values of y near 1 despite the fact that the p.d.f. of X is flat. Give an intuitive reason why this occurs in this example.
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