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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) ≤ 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

LetXbe a random variable with a continuous distribution.

Let \({{\bf{x}}_{\bf{1}}}{\bf{ = }}{{\bf{x}}_{\bf{2}}}{\bf{ = x}}\)

a.Prove that both \({{\bf{x}}_{\bf{1}}}\) and \({{\bf{x}}_{\bf{2}}}\) have a continuous distribution.

b.Prove that \({\bf{x = (}}{{\bf{x}}_{\bf{1}}}{\bf{,}}{{\bf{x}}_{\bf{2}}}{\bf{)}}\)does not have a continuous

joint distribution.

Question:Suppose that the joint p.d.f. ofXandYis as follows:

\(\)\({\bf{f}}\left( {{\bf{x,y}}} \right){\bf{ = }}\left\{ \begin{array}{l}\frac{{{\bf{15}}}}{{\bf{4}}}{{\bf{x}}^{\bf{2}}}\;{\bf{for}}\;{\bf{0}} \le {\bf{y}} \le {\bf{1 - }}{{\bf{x}}^{\bf{2}}}\\{\bf{0}}\;{\bf{otherwise}}\end{array} \right.\)

a. Determine the marginal p.d.f.’s ofXandY.

b. AreXandYindependent?

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.

In a large collection of coins, the probability X that a head will be obtained when a coin is tossed varies from one coin to another, and the distribution of X in the collection is specified by the following p.d.f.:

\({{\bf{f}}_{\bf{1}}}\left( {\bf{x}} \right){\bf{ = }}\left\{ {\begin{align}{}{{\bf{6x}}\left( {{\bf{1 - x}}} \right)}&{{\bf{for}}\,{\bf{0 < x < 1}}}\\{\bf{0}}&{{\bf{otherwise}}}\end{align}} \right.\)

Suppose that a coin is selected at random from the collection and tossed once, and that a head is obtained. Determine the conditional p.d.f. of X for this coin.

Suppose that \({{\bf{X}}_{\bf{1}}}{\bf{ \ldots }}{{\bf{X}}_{\bf{n}}}\) form a random sample of sizen from the uniform distribution on the interval [0, 1] andthat \({{\bf{Y}}_{\bf{n}}}{\bf{ = max}}\left( {{{\bf{X}}_{\bf{1}}}{\bf{ \ldots }}{{\bf{X}}_{\bf{n}}}} \right)\). Find the smallest value of \({\bf{n}}\)such that\({\bf{Pr}}\left( {{{\bf{Y}}_{\bf{n}}} \ge {\bf{0}}{\bf{.99}}} \right) \ge {\bf{0}}{\bf{.95}}\).
See all solutions

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