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Chapter 5: Q13E (page 280)

In a clinical trial with two treatment groups, the probability of success in one treatment group is 0.5, and the probability of success in the other is 0.6. Suppose that there are five patients in each group. Assume that the outcomes of all patients are independent. Calculate the probability that the first group will have at least as many successes as the second group.

Short Answer

Expert verified

The probability that the first group will have at least as many successes as the second group is 0.4957.

Step by step solution

01

Given information

The random variable X follows a binomial distribution, that is \(X \sim Bin\left( {n = 5,p} \right)\).

Two treatments are performed on two groups.

Group 1 has a probability of success that is p=0.5

Therefore, it follows, \(X \sim Bin\left( {n = 5,p = 0.5} \right)\)

Group 2 has a probability of success that is p=0.6

Therefore, it follows, \(X \sim Bin\left( {n = 5,p = 0.6} \right)\)

02

Formulate binomial probabilities table

03

Calculating the probability

The probability that the first group will have at least as many successes as the second group is:

\[0.03120{\rm{ }} \times 0.0102{\rm{ }} + {\rm{ }}0.1562{\rm{ }} \times {\rm{ }}\left( {0.0102{\rm{ }} + {\rm{ }}0.0768} \right){\rm{ }} + {\rm{ }}...{\rm{ }} + {\rm{ }}0.0312{\rm{ }} \times {\rm{ }}1.\]

The final answer is 0.4957.

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Most popular questions from this chapter

It is said that a random variable X has the Pareto distribution with parameters\({{\bf{x}}_{\bf{0}}}\,{\bf{and}}\,{\bf{\alpha }}\) if X has a continuous distribution for which the pdf\({\bf{f}}\left( {{\bf{x|}}\,{{\bf{x}}_{\bf{0}}}{\bf{,\alpha }}} \right)\) is as follows

\(\begin{array}{l}{\bf{f}}\left( {{\bf{x|}}\,{{\bf{x}}_{\bf{0}}}{\bf{,\alpha }}} \right){\bf{ = }}\frac{{{\bf{\alpha }}{{\bf{x}}_{\bf{0}}}^{\bf{\alpha }}}}{{{{\bf{x}}^{{\bf{\alpha + 1}}}}}}\,{\bf{,x}} \ge {{\bf{x}}_{\bf{0}}}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\bf{ = }}\,{\bf{0}}\,\,{\bf{,x < }}{{\bf{x}}_{\bf{0}}}\end{array}\)

Show that if X has this Pareto distribution, then the random variable\({\bf{log}}\left( {{\bf{X|}}\,{{\bf{x}}_{\bf{0}}}} \right)\)has the exponential distribution with parameter ฮฑ.

It is said that a random variableX has an increasing failure rate if the failure rate h(x) defined in Exercise 18 is an increasing function of xfor x> 0, and it is said that Xhas a decreasing failure rate if h(x) is a decreasing function of x for x > 0. Suppose that X has the Weibull distribution with parameters a and b, as defined in Exercise 19. Show thatX has an increasing failure rate if b > 1, and X has a decreasing failure rate if b < 1.

Let \({{\bf{X}}_{{\bf{1,}}}}{\bf{ }}{\bf{. }}{\bf{. }}{\bf{. , }}{{\bf{X}}_{\bf{n}}}\)be i.i.d. random variables having the normal distribution with mean \({\bf{\mu }}\) and variance\({{\bf{\sigma }}^{\bf{2}}}\). Define\(\overline {{{\bf{X}}_{\bf{n}}}} {\bf{ = }}\frac{{\bf{1}}}{{\bf{n}}}\sum\limits_{{\bf{i = 1}}}^{\bf{n}} {{{\bf{X}}_{\bf{i}}}} \) , the sample mean. In this problem, we shall find the conditional distribution of each \({{\bf{X}}_{\bf{i}}}\)given\(\overline {{{\bf{X}}_{\bf{n}}}} \).

a.Show that \({{\bf{X}}_{\bf{i}}}\)and\(\overline {{{\bf{X}}_{\bf{n}}}} \) have the bivariate normal distribution with both means \({\bf{\mu }}\), variances\({{\bf{\sigma }}^{\bf{2}}}{\rm{ }}{\bf{and}}\,\,\frac{{{{\bf{\sigma }}^{\bf{2}}}}}{{\bf{n}}}\),and correlation\(\frac{{\bf{1}}}{{\sqrt {\bf{n}} }}\).

Hint: Let\({\bf{Y = }}\sum\limits_{{\bf{j}} \ne {\bf{i}}} {{{\bf{X}}_{\bf{j}}}} \).

Now show that Y and \({{\bf{X}}_{\bf{i}}}\) are independent normal and \({{\bf{X}}_{\bf{n}}}\)and \({{\bf{X}}_{\bf{i}}}\) are linear combinations of Y and \({{\bf{X}}_{\bf{i}}}\) .

b.Show that the conditional distribution of \({{\bf{X}}_{\bf{i}}}\) given\(\overline {{{\bf{X}}_{\bf{n}}}} {\bf{ = }}\overline {{{\bf{x}}_{\bf{n}}}} \)\(\) is normal with mean \(\overline {{{\bf{x}}_{\bf{n}}}} \) and variance \({{\bf{\sigma }}^{\bf{2}}}\left( {{\bf{1 - }}\frac{{\bf{1}}}{{\bf{n}}}} \right)\).

Evaluate the integral\(\int\limits_0^\infty {{e^{ - 3{x^2}}}dx} \)

Suppose that an electronic system contains n components that function independently of each other, and suppose that these components are connected in series, as defined in Exercise 5 of Sec. 3.7. Suppose also that each component will function properly for a certain number of periods and then will fail. Finally, suppose that for i =1,...,n, the number of periods for which component i will function properly is a discrete random variable having a geometric distribution with parameter \({p_i}\). Determine the distribution of the number of periods for which the system will function properly.
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