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Chapter 3: Q19E (page 117)

Let X be a random variable with c.d.f. F and quantile function F-1. Let x0 and x1 be as defined in Exercise 17.

Short Answer

Expert verified

For all x in the open interval (x0,x1), F(x) is the largest p such thatF-1(p) 鈮� x

Step by step solution

01

Given information:

X be a random variable with cdf F and quantile function F-1

02

Step 2:Verifying F(x) is the largest p such that F-1 (p) ≤ x


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Most popular questions from this chapter

For the conditions of Exercise 1, find the p.d.f. of the

average \(\frac{{\left( {{{\bf{X}}_{\bf{1}}}{\bf{ + }}{{\bf{X}}_{\bf{2}}}} \right)}}{{\bf{2}}}\)

Suppose that thenrandom variablesX1. . . , Xnform arandom sample from a discrete distribution for which thep.f. is f. Determine the value of Pr(X1 = X2 = . . .= Xn).

Each student in a certain high school was classified according to her year in school (freshman, sophomore, junior, or senior) and according to the number of times that she had visited a certain museum (never, once, or more than once). The proportions of students in the various classifications are given in the following table:

Never once More than once

than once

Freshmen 0.08 0.10 0.04

Sophomores 0.04 0.10 0.04

Juniors 0.04 0.20 0.09

Seniors 0.02 0.15 0.10

a. If a student selected at random from the high school is a junior, what is the probability that she has never visited the museum?

b. If a student selected at random from the high school has visited the museum three times, what is the probability that she is a senior?

Question:Suppose thatXandYhave a continuous joint distribution for which the joint p.d.f. is

\({\bf{f}}\left( {{\bf{x,y}}} \right){\bf{ = }}\left\{ \begin{array}{l}{\bf{k}}\;{\bf{for}}\;{\bf{a}} \le {\bf{x}} \le {\bf{b}}\;{\bf{and}}\;{\bf{c}} \le {\bf{y}} \le {\bf{d}}\\{\bf{0}}\;{\bf{otherwise}}\end{array} \right.\)

wherea <b,c < d, andk >0.

Find the marginal distributions ofXandY.

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.
See all solutions

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