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Chapter 3: Q17E (page 117)

Prove that the quantile function F-1 of a general random variable X has the following three properties that are analogous to properties of the c.d.f.


Short Answer

Expert verified

a)

Step by step solution

01

verifying x0  is a non-decreasing function of p for 0 < p <1

a)

02

verifying p(X≤ c) > 0 and x1 equals the least upper bound on the set of the numbers d such that  p(X ≥  c) > 0

03

verifying is continuous from left

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Most popular questions from this chapter

Suppose that thenrandom variablesX1, . . . , Xnform a random sample from a continuous distribution for which the p.d.f. isf. Determine the probability that at leastk of thesenrandom variables will lie in a specified intervala鈮�x鈮�b.

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.

There are two boxes A and B, each containing red and green balls. Suppose that box A contains one red ball and two green balls and box B contains eight red balls and two green balls. Consider the following process: One ball is selected at random from box A, and one ball is selected at random from box B. The ball selected from box A is then placed in box B and the ball selected from box B is placed in box A. These operations are then repeated indefinitely. Show that the numbers of red balls in box A form a Markov chain with stationary transition probabilities, and construct the transition matrix of the Markov chain.

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 a random variableXhas the uniform distribution on the interval [鈭�2,8]. Find the p.d.f. ofXand the value of Pr(0<X <7).
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