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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

Question:Suppose that in a certain drug the concentration of aparticular chemical is a random variable with a continuousdistribution for which the p.d.f.gis as follows:

\({\bf{g}}\left( {\bf{x}} \right){\bf{ = }}\left\{ \begin{array}{l}\frac{{\bf{3}}}{{\bf{8}}}{{\bf{x}}^{\bf{2}}}\;{\bf{for}}\;{\bf{0}} \le {\bf{x}} \le {\bf{2}}\\{\bf{0}}\;{\bf{otherwise}}\end{array} \right.\)

Suppose that the concentrationsXandYof the chemicalin two separate batches of the drug are independent randomvariables for each of which the p.d.f. isg. Determine

(a) the joint p.d.f.of X andY;

(b) Pr(X=Y);

(c) Pr(X >Y );

(d) Pr(X+Y鈮�1).

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.

Prove Theorem 3.8.2. (Hint: Either apply Theorem3.8.4 or first compute the cdf. separately for a > 0 and a < 0.)

Consider the situation described in Example 3.7.14. Suppose that \(\) \({X_1} = 5\) and\({X_2} = 7\)are observed.

a. Compute the conditional p.d.f. of \({X_3}\) given \(\left( {{X_1},{X_2}} \right) = \left( {5,7} \right)\).

b. Find the conditional probability that \({X_3} > 3\)given \(\left( {{X_1},{X_2}} \right) = \left( {5,7} \right)\)and compare it to the value of \(P\left( {{X_3} > 3} \right)\)found in Example 3.7.9. Can you suggest a reason why the conditional probability should be higher than the marginal probability?

Two students,AandB,are both registered for a certain course. Assume that studentAattends class 80 percent of the time, studentBattends class 60 percent of the time, and the absences of the two students are independent. Consider the conditions of Exercise 7 of Sec. 2.2 again. If exactly one of the two students,AandB,is in class on a given day, what is the probability that it isA?
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