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Chapter 3: Q 11E (page 107)

Show that there does not exist any numbercsuch that the following functionf (x)would be a p.d.f.:

Short Answer

Expert verified

There does not exist any number c such that


is a probability density function.

Step by step solution

01

Given the information

The function is given as,


02

Calculate the value of c 

For a function to be a probability density function, the function should hold the following condition.

\int\limits_x^{}{f\left(x\right)}=1

For the provided function, the calculations are as follows:

Since, In (鈭� ) = 鈭� and In (0) is not defined, there does not exist any number c such that the provided function is a probability density function.

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Most popular questions from this chapter

Question:Suppose thatXandYhave a discrete joint distributionfor which the joint p.f. is defined as follows:

\({\bf{f}}\left( {{\bf{x,y}}} \right){\bf{ = }}\left\{ \begin{array}{l}\frac{{\bf{1}}}{{{\bf{30}}}}\left( {{\bf{x + y}}} \right)\;{\bf{for}}\;{\bf{x = 0,1,2}}\;{\bf{and}}\;{\bf{y = 0,1,2,3}}\\{\bf{0}}\;{\bf{otherwise}}\end{array} \right.\)

a. Determine the marginal p.f.鈥檚 ofXandY.

b. AreXandYindependent?

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 the joint p.d.f. of two random variables X and Y is as follows:

\(f\left( {x,y} \right) = \left\{ \begin{aligned}{l}c\left( {x + {y^2}} \right)\,\,\,\,\,\,for\,0 \le x \le 1\,and\,0 \le y \le 1\\0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,otherwise\end{aligned} \right.\)

Determine

(a) the conditional p.d.f. of X for every given value of Y, and

(b) \({\rm P}\left( {X > \frac{1}{2}|Y = \frac{3}{2}} \right)\).

Suppose that a random variableXhas the binomial distribution with parametersn=15 andp=0.5. Find Pr(X <6).

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?
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