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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

Let Y be the rate (calls per hour) at which calls arrive at a switchboard. Let X be the number of calls during a two-hour period. Suppose that the marginal p.d.f. of Y is

\({{\bf{f}}_{\bf{2}}}\left( {\bf{y}} \right){\bf{ = }}\left\{ {\begin{align}{}{{{\bf{e}}^{{\bf{ - y}}}}}&{{\bf{if}}\,{\bf{y > 0,}}}\\{\bf{0}}&{{\bf{otherwise,}}}\end{align}} \right.\)

And that the conditional p.d.f. of X given\({\bf{Y = y}}\)is

\({{\bf{g}}_{\bf{1}}}\left( {{\bf{x}}\left| {\bf{y}} \right.} \right){\bf{ = }}\left\{ {\begin{align}{}{\frac{{{{\left( {{\bf{2y}}} \right)}^{\bf{x}}}}}{{{\bf{x!}}}}{{\bf{e}}^{{\bf{ - 2y}}}}}&{{\bf{if}}\,{\bf{x = 0,1,}}...{\bf{,}}}\\{\bf{0}}&{{\bf{otherwise}}{\bf{.}}}\end{align}} \right.\)

  1. Find the marginal p.d.f. of X. (You may use the formula\(\int_{\bf{0}}^\infty {{{\bf{y}}^{\bf{k}}}{{\bf{e}}^{{\bf{ - y}}}}{\bf{dy = k!}}} {\bf{.}}\))
  2. Find the conditional p.d.f.\({{\bf{g}}_{\bf{2}}}\left( {{\bf{y}}\left| {\bf{0}} \right.} \right)\)of Y given\({\bf{X = 0}}\).
  3. Find the conditional p.d.f.\({{\bf{g}}_{\bf{2}}}\left( {{\bf{y}}\left| 1 \right.} \right)\)of Y given\({\bf{X = 1}}\).
  4. For what values of y is\({{\bf{g}}_{\bf{2}}}\left( {{\bf{y}}\left| {\bf{1}} \right.} \right){\bf{ > }}{{\bf{g}}_{\bf{2}}}\left( {{\bf{y}}\left| {\bf{0}} \right.} \right)\)? Does this agree with the intuition that the more calls you see, the higher you should think the rate is?

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.

Let Xbe a random variable for which the p.d.f. is as in Exercise 5. After the value ofXhas been observed, letYbe the integer closest toX. Find the p.f. of the random variableY.

Suppose that a random variableXhas the uniform distributionon the integers 10, . . . ,20. Find the probability thatXis even.

Suppose that\({{\bf{X}}_{\bf{1}}}\)and\({{\bf{X}}_{\bf{2}}}\)are i.i.d. random variables, and that each has the uniform distribution on the interval[0,1]. Evaluate\({\bf{P}}\left( {{{\bf{X}}_{\bf{1}}}^{\bf{2}}{\bf{ + }}{{\bf{X}}_{\bf{2}}}^{\bf{2}} \le {\bf{1}}} \right)\)
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