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Chapter 3: Q1E (page 129)

Suppose that the joint p.d.f. of a pair of random variables (X,Y) is constant on the rectangle where 0鈮鈮�2 and 0鈮� y 鈮� 1, and suppose that the p.d.f. is 0 off of this rectangle.

a. Find the constant value of the p.d.f. on the rectangle.

b. Find Pr (X鈮)

Short Answer

Expert verified

a. The constant value is 0.50

b. The probability is 0.75

Step by step solution

01

Given information

The pdf of X is given by,


02

Finding the value of constant

03

Calculating the probability

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Most popular questions from this chapter

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 thatXandYare random variables such that(X, Y)must belong to the rectangle in thexy-plane containing all points(x, y)for which 0鈮�x鈮�3 and 0鈮�y鈮�4. Suppose also that the joint c.d.f. ofXandYat every point

(x,y) in this rectangle is specified as follows:

\({\bf{F}}\left( {{\bf{x,y}}} \right){\bf{ = }}\frac{{\bf{1}}}{{{\bf{156}}}}{\bf{xy}}\left( {{{\bf{x}}^{\bf{2}}}{\bf{ + y}}} \right)\)

Determine

(a) Pr(1鈮�X鈮�2 and 1鈮�Y鈮�2);

(b) Pr(2鈮�X鈮�4 and 2鈮�Y鈮�4);

(c) the c.d.f. ofY;

(d) the joint p.d.f. ofXandY;

(e) Pr(Y鈮�X).

Suppose that a random variableXhas a discrete distribution

with the following p.f.:

\(f\left( x \right) = \left\{ \begin{array}{l}\frac{c}{{{2^x}}}\;\;for\;x = 0,1,2,...\\0\;\;\;\;otherwise\end{array} \right.\)

Find the value of the constantc.

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).

Question:A painting process consists of two stages. In the first stage, the paint is applied, and in the second stage, a protective coat is added. Let X be the time spent on the first stage, and let Y be the time spent on the second stage. The first stage involves an inspection. If the paint fails the inspection, one must wait three minutes and apply the paint again. After a second application, there is no further inspection. The joint pdf.of X and Y is

\(f\left( {x,y} \right) = \left\{ \begin{array}{l}\frac{1}{3}if\,1 < x < 3\,and\,0 < y < 1\\\frac{1}{6}if\,1 < x < 3\,and\,0 < y < 1\,\\0\,\,otherwise.\\\,\end{array} \right.\,\,\)

a. Sketch the region where f (x, y) > 0. Note that it is not exactly a rectangle.

b. Find the marginal p.d.f.鈥檚 of X and Y.

c. Show that X and Y are independent.
See all solutions

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