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Chapter 3: Q7E (page 117)

Suppose, as in Exercise 7 of Sec. 3.2, that a random variable X has a uniform distribution on the interval [鈭�2, 8]. Find and sketch the c.d.f. of X.

Short Answer

Expert verified

A random variable X has a uniform distribution on the interval [鈭�2, 8]

Step by step solution

01

Given information

A random variable X has a uniform distribution on the interval [鈭�2, 8]

02

Finding the probability density function

The pdf of uniform distribution is given by:


03

Derivation of the distribution function.

04

Sketch the CDF

X-axis contain the values of the random variable X and Y-axis contains its corresponding CDF values.

Plotting values of X and Y we get the following graph.

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Most popular questions from this chapter

Question:Consider the clinical trial of depression drugs in Example2.1.4. Suppose that a patient is selected at random from the 150 patients in that study and we recordY, an indicator of the treatment group for that patient, andX, an indicator of whether or not the patient relapsed. Table 3.3contains the joint p.f. ofXandY.

Response(X)

Treatment Group(Y)

Impramine(1)

Lithium(2)

Combination(3)

Placebo(4)

Relapse(0)

0.120

0.087

0.146

0.160

No relapse(1)

0.147

0.166

0.107

0.067

a. Calculate the probability that a patient selected at random from this study used Lithium (either alone or in combination with Imipramine) and did not relapse.

b. Calculate the probability that the patient had a relapse(without regard to the treatment group).

Suppose that a person鈥檚 score X on a mathematics aptitude test is a number between 0 and 1, and that his score Y on a music aptitude test is also a number between 0 and 1. Suppose further that in the population of all college students in the United States, the scores X and Y are distributed according to the following joint pdf:

\(f\left( {x,y} \right)\left\{ \begin{aligned}\frac{2}{5}\left( {2x + 3y} \right)for0 \le x \le 1 and 0 \le y \le 1\\0 otherwise\end{aligned} \right.\)

a. What proportion of college students obtain a score greater than 0.8 on the mathematics test?

b. If a student鈥檚 score on the music test is 0.3, what is the probability that his score on the mathematics test will be greater than 0.8?

c. If a student鈥檚 score on the mathematics test is 0.3, what is the probability that his score on the music test will be greater than 0.8?

Let Xbe a random variable for which the p.d.f. is as in Exercise 5. After the value ofXhas been observed, letYbe the integer closest toX. Find the p.f. of the random variableY.

Let X have the uniform distribution on the interval, and let prove that \({\bf{cX + d}}\) it has a uniform distribution on the interval \(\left[ {{\bf{ca + d,cb + d}}} \right]\)

Suppose that a Markov chain has four states 1, 2, 3, 4, and stationary transition probabilities as specified by the following transition matrix

\(p = \left[ {\begin{array}{*{20}{c}}{\frac{1}{4}}&{\frac{1}{4}}&0&{\frac{1}{2}}\\0&1&0&0\\{\frac{1}{2}}&0&{\frac{1}{2}}&0\\{\frac{1}{4}}&{\frac{1}{4}}&{\frac{1}{4}}&{\frac{1}{4}}\end{array}} \right]\):

a.If the chain is in state 3 at a given timen, what is the probability that it will be in state 2 at timen+2?

b.If the chain is in state 1 at a given timen, what is the probability it will be in state 3 at timen+3?
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