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Chapter 3: Q 6E (page 167)

Suppose that thenrandom variablesX1. . . , Xnform arandom sample from a discrete distribution for which thep.f. is f. Determine the value of Pr(X1 = X2 = . . .= Xn).

Short Answer

Expert verified

The value of \(pr\left( {{x_1}, \ldots ,{x_n}} \right)\) is \({p^{\sum\limits_{i = 1}^n {{x_i}} }}{(1 - p)^{n - \sum\limits_{i = 1}^n {{x_i}} }}\).

Step by step solution

01

Given Information

Here given distribution is discrete distribution for \(n\) random variables.

02

State the random variables

For \(n\) random variables we take let \({x_1}, \ldots ,{x_n} \sim Ber\left( p \right)\) . Here we assume that the random variables follow Bernoulli distribution.

03

Step 3:Compute the pmf value

For discrete distribution to compute the pmf is

\(\begin{align}pr\left( {{x_1}, \ldots ,{x_n}} \right)\\ &= f\left( {{x_1}, \ldots ,{x_n}} \right)\\ &= {p^{{x_1} + , \ldots , + {x_n}}}{\left( {1 - p} \right)^{n - \left( {{x_1} + , \ldots , + {x_n}} \right)}}\\ &= {p^{\sum\limits_{i = 1}^n {{x_i}} }}{(1 - p)^{n - \sum\limits_{i = 1}^n {{x_i}} }}\end{align}\)

Hence determine the value is\({p^{\sum\limits_{i = 1}^n {{x_i}} }}{(1 - p)^{n - \sum\limits_{i = 1}^n {{x_i}} }}\).

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Most popular questions from this chapter

Suppose that a random variableXhas the uniform distribution on the interval [鈭�2,8]. Find the p.d.f. ofXand the value of Pr(0<X <7).

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\({\bf{Pr}}\left( {{{\bf{Y}}_{\bf{n}}} \ge {\bf{0}}{\bf{.99}}} \right) \ge {\bf{0}}{\bf{.95}}\).

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?

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

\(\begin{aligned}f\left( x \right) &= e{}^{ - x},x > 0\\ &= 0,x \le 0\end{aligned}\)

Determine the p.d.f. of \({\bf{Y = }}{{\bf{X}}^{\frac{{\bf{1}}}{{\bf{2}}}}}\)

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?
See all solutions

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