8.2 Problems on random vectors and joint distributions (Page 14/6)

Applied probability Page 1 / 6

Two cards are selected at random, without replacement, from a standard deck. Let X be the number of aces and Y be the number of spades. Under the usual assumptions, determine the joint distribution and the marginals.

Let X be the number of aces and Y be the number of spades. Define the events $A S_{i}$ , A _i , S _i , and N _i , $i = 1, 2$ , of drawing ace of spades, other ace, spade (other than the ace), and neither on the i selection. Let $P (i, k) = P (X = i, Y = k)$ .

$P (0, 0) = P (N_{1} N_{2}) = \frac{36}{52} • \frac{35}{51} = \frac{1260}{2652}$

$P (0, 1) = P (N_{1} S_{2} ⋁ S_{1} N_{2}) = \frac{36}{52} • \frac{12}{51} + \frac{12}{52} • \frac{36}{51} = \frac{864}{2652}$

$P (0, 2) = P (S_{1} S_{2}) = \frac{12}{52} • \frac{11}{51} = \frac{132}{2652}$

$P (1, 0) = P (A_{1} N_{2} ⋁ N_{1} S_{2}) = \frac{3}{52} • \frac{36}{51} + \frac{36}{52} • \frac{3}{51} = \frac{216}{2652}$

$P (1, 1) = P (A_{1} S_{2} ⋁ S_{1} A_{2} ⋁ A S_{1} N_{2} ⋁ N_{1} A S_{2}) = \frac{3}{52} • \frac{12}{51} + \frac{12}{52} • \frac{3}{51} + \frac{1}{52} • \frac{36}{51} + \frac{36}{52} • \frac{1}{51} = \frac{144}{2652}$

$P (1, 2) = P (A S_{1} S_{2} ⋁ S_{1} A S_{2}) = \frac{1}{52} • \frac{12}{51} + \frac{12}{52} • \frac{1}{51} = \frac{24}{2652}$

$P (2, 0) = P (A_{1} A_{2}) = \frac{3}{52} • \frac{2}{51} = \frac{6}{2652}$

$P (2, 1) = P (A S_{1} A_{2} ⋁ A_{1} A S_{2}) = \frac{1}{52} • \frac{3}{51} + \frac{3}{52} • \frac{1}{51} = \frac{6}{2652}$

$P (2, 2) = P (\emptyset) = 0$

% type
npr08_01 % file
npr08_01.m % Solution for
[link] X = 0:2;
Y = 0:2;Pn = [132  24   0; 864 144  6; 1260 216 6];P = Pn/(52*51);
disp('Data in Pn, P, X, Y')npr08_01         % Call for mfileData in Pn, P, X, Y    % Result
PX = sum(P)PX =  0.8507    0.1448    0.0045
PY = fliplr(sum(P'))PY =  0.5588    0.3824    0.0588

Got questions? Get instant answers now!

Two positions for campus jobs are open. Two sophomores, three juniors, and three seniors apply. It is decided to select two at random (each possible pairequally likely). Let X be the number of sophomores and Y be the number of juniors who are selected. Determine the joint distribution for the pair ${X, Y}$ and from this determine the marginals for each.

Let $A_{i}, B_{i}, C_{i}$ be the events of selecting a sophomore, junior, or senior, respectively, on the i th trial. Let X be the number of sophomores and Y be the number of juniors selected.

Set $P (i, k) = P (X = i, Y = k)$

$P (0, 0) = P (C_{1} C_{2}) = \frac{3}{8} • \frac{2}{7} = \frac{6}{56}$

$P (0, 1) = P (B_{1} C_{2}) + P (C_{1} B_{2}) = \frac{3}{8} • \frac{3}{7} + \frac{3}{8} • \frac{3}{7} = \frac{18}{56}$

$P (0, 2) = P (B_{1} B_{2}) = \frac{3}{8} • \frac{2}{7} = \frac{6}{56}$

$P (1, 0) = P (A_{1} C_{2}) + P (C_{1} A_{2}) = \frac{2}{8} • \frac{3}{7} + \frac{3}{8} • \frac{2}{7} = \frac{12}{56}$

$P (1, 1) = P (A_{1} B_{2}) + P (B_{1} A_{2}) = \frac{2}{8} • \frac{3}{7} + \frac{3}{8} • \frac{2}{7} = \frac{12}{56}$

$P (2, 0) = P (A_{1} A_{2}) = \frac{2}{8} • \frac{1}{7} = \frac{2}{56}$

$P (1, 2) = P (2, 1) = P (2, 2) = 0$

$P X = [30 / 56 24 / 56 2 / 56] P Y = [20 / 56 30 / 56 6 / 56]$

% file
npr08_02.m % Solution for
[link] X = 0:2;
Y = 0:2;Pn = [6 0 0; 18 12 0; 6 12 2];P = Pn/56;
disp('Data are in X, Y,Pn, P')
npr08_02 Data are in X, Y,Pn, P
PX = sum(P)PX =  0.5357    0.4286    0.0357
PY = fliplr(sum(P'))PY =  0.3571    0.5357    0.1071

Got questions? Get instant answers now!

A die is rolled. Let X be the number that turns up. A coin is flipped X times. Let Y be the number of heads that turn up. Determine the joint distribution for the pair ${X, Y}$ . Assume $P (X = k) = 1 / 6$ for $1 \leq k \leq 6$ and for each k , $P (Y = j | X = k)$ has the binomial $(k, 1 / 2)$ distribution. Arrange the joint matrix as on the plane, with values of Y increasing upward. Determine the marginal distribution for Y . (For a MATLAB based way to determine the joint distribution see Example 7 from "Conditional Expectation, Regression")

$P (X = i, Y = k) = P (X = i) P (Y = k | X = i) = (1 / 6) P (Y = k | X = i)$ .

% file
npr08_03.m % Solution for
[link] X = 1:6;
Y = 0:6;P0 = zeros(6,7);       % Initialize
for i = 1:6            % Calculate rows of Y probabilitiesP0(i,1:i+1) = (1/6)*ibinom(i,1/2,0:i);
endP = rot90(P0);         % Rotate to orient as on the plane
PY = fliplr(sum(P'));  % Reverse to put in normal orderdisp('Answers are in X, Y, P, PY')
npr08_03 % Call for solution m-file
Answers are in X, Y, P, PYdisp(P)
         0         0         0         0         0    0.00260         0         0         0    0.0052    0.0156
         0         0         0    0.0104    0.0260    0.03910         0    0.0208    0.0417    0.0521    0.0521
         0    0.0417    0.0625    0.0625    0.0521    0.03910.0833    0.0833    0.0625    0.0417    0.0260    0.0156
    0.0833    0.0417    0.0208    0.0104    0.0052    0.0026disp(PY)
     0.1641  0.3125  0.2578  0.1667  0.0755  0.0208  0.0026

Got questions? Get instant answers now!

Questions & Answers

what is defense mechanism

Chinaza Reply

what is defense mechanisms

Chinaza

I'm interested in biological psychology and cognitive psychology

Tanya Reply

what does preconceived mean

sammie Reply

physiological Psychology

Nwosu Reply

How can I develope my cognitive domain

Amanyire Reply

why is communication effective

Dakolo Reply

Communication is effective because it allows individuals to share ideas, thoughts, and information with others.

effective communication can lead to improved outcomes in various settings, including personal relationships, business environments, and educational settings. By communicating effectively, individuals can negotiate effectively, solve problems collaboratively, and work towards common goals.

it starts up serve and return practice/assessments.it helps find voice talking therapy also assessments through relaxed conversation.

miss

Every time someone flushes a toilet in the apartment building, the person begins to jumb back automatically after hearing the flush, before the water temperature changes. Identify the types of learning, if it is classical conditioning identify the NS, UCS, CS and CR. If it is operant conditioning, identify the type of consequence positive reinforcement, negative reinforcement or punishment

Wekolamo Reply

please i need answer

Wekolamo

because it helps many people around the world to understand how to interact with other people and understand them well, for example at work (job).

Manix Reply

Agreed 👍 There are many parts of our brains and behaviors, we really need to get to know. Blessings for everyone and happy Sunday!

ARC

A child is a member of community not society elucidate ?

JESSY Reply

Isn't practices worldwide, be it psychology, be it science. isn't much just a false belief of control over something the mind cannot truly comprehend?

Simon Reply

compare and contrast skinner's perspective on personality development on freud

namakula Reply

Skinner skipped the whole unconscious phenomenon and rather emphasized on classical conditioning

war

explain how nature and nurture affect the development and later the productivity of an individual.

Amesalu Reply

nature is an hereditary factor while nurture is an environmental factor which constitute an individual personality. so if an individual's parent has a deviant behavior and was also brought up in an deviant environment, observation of the behavior and the inborn trait we make the individual deviant.

Samuel

I am taking this course because I am hoping that I could somehow learn more about my chosen field of interest and due to the fact that being a PsyD really ignites my passion as an individual the more I hope to learn about developing and literally explore the complexity of my critical thinking skills

Zyryn Reply

good👍

Jonathan

and having a good philosophy of the world is like a sandwich and a peanut butter 👍

Jonathan

generally amnesi how long yrs memory loss

Kelu Reply

interpersonal relationships

Abdulfatai Reply

Got questions? Join the online conversation and get instant answers!

Jobilize.com Reply

<< Chapter < Page Page > Chapter >>

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

100% Free Mobile Applications
Receive real-time job alerts and never miss the right job again

Source: OpenStax, Applied probability. OpenStax CNX. Aug 31, 2009 Download for free at http://cnx.org/content/col10708/1.6

Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Applied probability' conversation and receive update notifications?

Ask

	8 AP 08 Appendicular Skeleton MCQ By OpenStax Start Quiz
	18 AP Key Terms 18 Cardiovascular System Blood By OpenStax Start Key Terms
	24 PLANT QUIZ 1 By Brooke Delaney Start Exam
©flickr: Jovial	Video Mid Term By Dewey Compton Start Exam
	35 Biology 35 The Nervous System MCQ By OpenStax Start Quiz
	10 Lec:10 Sampling Confidence Intervals By Janet Forrester Start Quiz
	9 Neuroanatomy 09 The Auditory System By Stephen Voron Start Quiz
	15 Biology 15 Genes and Proteins MCQ By OpenStax Start Quiz
	4 BOD Hemolymphatic -Dr. Han By Brooke Delaney Start Exam
	26 AP 26 Fluid, Electrolyte, Acid-Base Balance MCQ By OpenStax Start Quiz