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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≤x≤2 and 0≤ y ≤ 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≥Y)

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\({X_1}...{X_n}\)are independent. Let\(k < n\)and let\({i_1}.....{i_k}\)be distinct integers between 1 and n. Prove that \(X{i_1}.....X{i_k}\)they are independent.

Start with the joint distribution of treatment group and response in Table 3.6 on page 138. For each treatment group, compute the conditional distribution of response given the treatment group. Do they appear to be very similar or quite different?

Suppose that the joint distribution of X and Y is uniform over a set A in the xy-plane. For which of the following sets A are X and Y independent?

a. A circle with a radius of 1 and with its center at the origin

b. A circle with a radius of 1 and with its center at the point (3,5)

c. A square with vertices at the four points (1,1), (1,−1), (−1,−1), and (−1,1)

d. A rectangle with vertices at the four points (0,0), (0,3), (1,3), and (1,0)

e. A square with vertices at the four points (0,0), (1,1),(0,2), and (−1,1)

Question:A certain drugstore has three public telephone booths. Fori=0, 1, 2, 3, let\({{\bf{p}}_{\bf{i}}}\)denote the probability that exactlyitelephone booths will be occupied on any Monday evening at 8:00 p.m.; and suppose that\({{\bf{p}}_{\bf{0}}}\)=0.1,\({{\bf{p}}_{\bf{1}}}\)=0.2,\({{\bf{p}}_{\bf{2}}}\)=0.4, and\({{\bf{p}}_{\bf{3}}}\)=0.3. LetXandYdenote the number of booths that will be occupied at 8:00 p.m. on two independent Monday evenings. Determine:

(a) the joint p.f. ofXandY;

(b) Pr(X=Y);

(c) Pr(X > Y ).

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.’s ofXandY.

b. AreXandYindependent?
See all solutions

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