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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 \({{\bf{X}}_{\bf{1}}}\;{\bf{and}}\;{{\bf{X}}_{\bf{2}}}\)are i.i.d. random variables andthat the p.d.f. of each of them is as follows:

\({\bf{f}}\left( {\bf{x}} \right){\bf{ = }}\left\{ \begin{array}{l}{{\bf{e}}^{{\bf{ - x}}}}\;\;\;\;\;\;{\bf{for}}\;{\bf{x > 0}}\\{\bf{0}}\;\;\;\;\;\;\;\;{\bf{otherwise}}\end{array} \right.\)

Find the p.d.f. of \({\bf{Y = }}{{\bf{X}}_{\bf{1}}} - {{\bf{X}}_{\bf{2}}}\)

Suppose that a random variableXhas a discrete distribution

with the following p.f.:

\(f\left( x \right) = \left\{ \begin{array}{l}cx\;\;for\;x = 1,...,5,\\0\;\;\;\;otherwise\end{array} \right.\)

Determine the value of the constantc.

Let X and Y be random variables for which the jointp.d.f. is as follows:

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

Find the p.d.f. of Z = X + Y.

Question:Suppose that the joint p.d.f. of X and Y is as follows:

\(f\left( {x,y} \right) = \left\{ \begin{array}{l}24xy for x \ge 0,y \ge 0, and x + y \le 1,\\0 otherwise\end{array} \right.\).

Are X and Y independent?

A civil engineer is studying a left-turn lane that is long enough to hold seven cars. LetXbe the number of cars in the lane at the end of a randomly chosen red light. The engineer believes that the probability thatX=xis proportional to(x+1)(8−x)forx=0, . . . ,7 (the possible values ofX).

a. Find the p.f. ofX.

b. Find the probability thatXwill be at least 5.
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