240 Bivariate Probability Distributions and Sampling D Chapter 6 the behavior of

Question

240 Bivariate Probability Distributions and Sampling D Chapter 6 the behavior of granular media. Refer to the Engineering Computations: International Journal for Computer-Aided Engineering and Sofhevare (Vol. 30, No 2 2013) study of the properties of granular media (e.g 6.6 sand, rice, ball bearings, and flour), Exercise 3.62 (p. 120). The study assumes there is a system of N non-interacting granular particles, where the particles are grouped accord- ing to energy level. r. For this problem (as in Exercise 3.62), assume that N = 7 and r = 3, then consider the scenario where there is one particle (of the total of 7 parti- cles) at energy level 1, two particles at energy level 2, and four particles at energy level 3. Another feature of the par- ticles studied was the position in time where the particle reached a certain entropy level during compression. All particles reached the desired entropy level at one of three time periods, 1. 2. or 3. Assume the 7 particles had the characteristics shown in the table. Consider a randomly selected particle and let X represent the energy level and Y the time period associated with particle Particle iD Energy Level Time Period 6.7 the bivariate probability distribution, p[x. y). b Find the marginal distribution, pi(x) c. Find the marginal distribution, pó). d. Find the conditional distribution., paly x). 6S Variable speed limit control for freewas Rafau

Explanation / Answer

(a)

Here X can take values 1, 2, 3. And Y can also take values 1, 2, and 3.

Out of 7 particles, one particle has X=1 and Y=1 so

P(X=1, Y=1) = 1/7

Out of 7 particles, two particles have X=2 and Y=1 so

P(X=2, Y=1) = 2/7

Likewise folloiwng is the complete joint probability distribution table:

(b)

The marginal pdf of X is shown by column total of joint probability table. Following table shows the marginal pdf:

(c)

The marginal pdf of Y is shown by row total of joint probability table. Following table shows the marginal pdf:

(d)

The conditional pdf of Y given X =x, x=1,2,3 is:

X 1 2 3 P(Y=y) 1 1/7 0 0 1/7 Y 2 2/7 0 0 2/7 3 1/7 2/7 1/7 4/7 P(X=x) 4/7 2/7 1/7 1

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