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Chapter 3: Q10E (page 107)

Suppose that the p.d.f. of a random variable X is as follows:

f(x) = {c/(1-x)1/2 for 0 <x< 1,

0 otherwise.

a. Find the value of the constant c and sketch the p.d.f.

b. Find the value of Pr(X ≤ 1/2).

Short Answer

Expert verified

The value is 0.2928

Step by step solution

01

Given information

02

(a) Find the value of the constant c and draw the sketchStep-by-step solution

Put 1-x=t

-dx=dt

X=0 and t=1

X=1 and t=0

03

b) Find the value of the Pr (X ≤ 1/2).

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Most popular questions from this chapter

Suppose that a box contains seven red balls and three blue balls. If five balls are selected at random, without replacement, determine the p.f. of the number of red balls that will be obtained.

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 a coin is tossed repeatedly in such a way that heads and tails are equally likely to appear on any given toss and that all tosses are independent, with the following exception: Whenever either three heads or three tails have been obtained on three successive tosses, then the outcome of the next toss is always of the opposite type. At time\(n\left( {n \ge 3} \right)\)let the state of this process be specified by the outcomes on tosses\(n - 2\),\(n - 1\)and n. Show that this process is a Markov chain with stationary transition probabilities and construct the transition matrix.

Find the unique stationary distribution for the Markov chain in Exercise 2.

Suppose that either of two instruments might be used for making a certain measurement. Instrument 1 yields a measurement whose p.d.f.\({{\bf{h}}_{\bf{1}}}\)is

\({{\bf{h}}_{\bf{1}}}\left( {\bf{x}} \right){\bf{ = }}\left\{ {\begin{align}{}{{\bf{2x}}}&{{\bf{for}}\,{\bf{0 < x < 1}}}\\{\bf{0}}&{{\bf{otherwise}}}\end{align}} \right.\)

Instrument 2 yields a measurement whose p.d.f.\({{\bf{h}}_2}\)is

\({{\bf{h}}_{\bf{2}}}\left( {\bf{x}} \right){\bf{ = }}\left\{ {\begin{align}{}{{\bf{3}}{{\bf{x}}^{\bf{2}}}}&{{\bf{for}}\,{\bf{0 < x < 1}}}\\{\bf{0}}&{{\bf{otherwise}}}\end{align}} \right.\)

Suppose that one of the two instruments is chosen randomly, and a measurement X is made with it.

  1. Determine the marginal p.d.f. of X.
  2. If the measurement value is\({\bf{X = }}{\raise0.7ex\hbox{\({\bf{1}}\)} \!\mathord{\left/ {\vphantom {{\bf{1}} {\bf{4}}}}\right.\ } \!\lower0.7ex\hbox{\({\bf{4}}\)}}\), what is the probability that instrument 1 was used?
See all solutions

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