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Chapter 1: Q10E (page 1)

Question:Suppose that a point (X, Y ) is chosen at random from the circle S defined as follows:\({\rm{S}} = \left\{ {\left( {x,y} \right):{x^2} + {y^2} \le 1} \right\}\)

a. Determine the joint p.d.f. of X and Y , the marginal p.d.f. of X, and the marginal p.d.f. of Y .

b. Are X and Y independent

Short Answer

Expert verified

\(f\left( {x,y} \right)\)is constant over the circle S.

Area of S=\(\pi \) unit

Step by step solution

01

Given information

\(f\left( {x,y} \right)\)is constant over the circle S.

Area of S=\(\pi \) unit

02

Calculating pfd and marginal pdf.

a.

Area of s is\(\pi \)

By symmetry, the random variable Y will have the same marginal pdf as x:

03

Checking for independency of variables.

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Most popular questions from this chapter

Suppose that 35 people are divided in a random manner into two teams in such a way that one team contains10 people and the other team contains 25 people. What is the probability that two particular people A and B will be on the same team?

Each minute a machine produces a length of rope with mean of 4 feet and standard deviation of 5 inches. Assuming that the amounts produced in different minutes are independent and identically distributed, approximate the probability that the machine will produce at least 250 feet in one hour.

Consider a state lottery game in which each winning combination and each ticket consists of one set of k numbers chosen from the numbers 1 to n without replacement. We shall compute the probability that the winning combination contains at least one pair of consecutive numbers.

a. Prove that if\({\bf{n < 2k - 1}}\), then every winning combination has at least one pair of consecutive numbers. For the rest of the problem, assume that\({\bf{n}} \le {\bf{2k - 1}}\).

b. Let\({{\bf{i}}_{\bf{1}}}{\bf{ < }}...{\bf{ < }}{{\bf{i}}_{\bf{k}}}\)be an arbitrary possible winning combination arranged in order from smallest to largest. For\({\bf{s = 1,}}...{\bf{,k}}\), let\({{\bf{j}}_{\bf{s}}}{\bf{ = }}{{\bf{i}}_{\bf{s}}}{\bf{ - }}\left( {{\bf{s - 1}}} \right)\). That is,

\(\begin{array}{c}{{\bf{j}}_{\bf{1}}}{\bf{ = }}{{\bf{i}}_{\bf{1}}}\\{{\bf{j}}_{\bf{2}}}{\bf{ = }}{{\bf{i}}_{\bf{2}}}{\bf{ - 1}}\\{\bf{.}}\\{\bf{.}}\\{\bf{.}}\\{{\bf{j}}_{\bf{k}}}{\bf{ = }}{{\bf{i}}_{\bf{k}}}{\bf{ - }}\left( {{\bf{k - 1}}} \right)\end{array}\)

Prove that\(\left( {{{\bf{i}}_{\bf{1}}}{\bf{,}}...{\bf{,}}{{\bf{i}}_{\bf{k}}}} \right)\)contains at least one pair of consecutive numbers if and only if\(\left( {{{\bf{j}}_{\bf{1}}}{\bf{,}}...{\bf{,}}{{\bf{j}}_{\bf{k}}}} \right)\)contains repeated numbers.

c. Prove that\({\bf{1}} \le {{\bf{j}}_{\bf{1}}} \le ... \le {{\bf{j}}_{\bf{k}}} \le {\bf{n - k + 1}}\)and that the number of\(\left( {{{\bf{j}}_{\bf{1}}}{\bf{,}}...{\bf{,}}{{\bf{j}}_{\bf{k}}}} \right)\)sets with no repeats is\(\left( {\begin{array}{*{20}{c}}{{\bf{n - k + 1}}}\\{\bf{k}}\end{array}} \right)\)

d. Find the probability that there is no pair of consecutive numbers in the winning combination.

e. Find the probability of at least one pair of consecutive numbers in the winning combination

Suppose that n letters are placed at random in n envelopes, and let \({q_n}\) denote the probability that no letter is placed in the correct envelope. For which of the following four values of n is \({q_n}\)largest: n = 10, n = 21, n = 53, or n = 300?

Suppose that X is a random variable for which\({\bf{P}}\left( {{\bf{X}} \ge {\bf{0}}} \right){\bf{ = 1}}\,{\bf{and}}\,{\bf{P}}\left( {{\bf{X}} \ge {\bf{10}}} \right){\bf{ = }}\frac{{\bf{1}}}{{\bf{5}}}\) .

Prove that \({\bf{E}}\left( {\bf{X}} \right) \ge {\bf{2}}\) .
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