Mathematics > EXAM REVIEW > Solutions Manual > Georgia Institute Of Technology CS 7641 CSE/ISYE 6740 Mid-term Exam 2 (Fall 2016) (All)

Solutions Manual > Georgia Institute Of Technology CS 7641 CSE/ISYE 6740 Mid-term Exam 2 (Fall 2016) Solutions/ Fall 2016 Midterm2

Document Content and Description Below

Georgia Institute Of Technology CS 7641 CSE/ISYE 6740 Mid-term Exam 2 (Fall 2016) Solutions Probability and Bayes’ Rule [14 pts] (a) A probability density function (pdf) is defined by f(x; y) =... (C 0 otherwise (x + 2y) if 0 < y < 1 and 0 < x < 2 (i) Find the value of C [3 pts]. Answer: Z01 Z02 C(x + 2y)dxdy = 4C = 1 Thus, C = 1 4: (ii) Find the marginal distribution of X [2 pts]. Answer: fX(x) = (R 0 otherwise 01 1 4(x + 2y)dy = 14(x + 1) if 0 < x < 2 (iii) Find the joint cumulative density function (cdf) of X and Y [2 pts]. Answer: The complete definition of FXY is: FXY (x; y) = 8>>>>>><>>>>>>: 0 | x ≤ 0 or y ≤ 0 x2y=8 + y2x=4 0 < x < 2 and 0 < y < 1 y=2 + y2=2 x2=8 + x=4 | 2 ≤ x and 0 < y < 1 0 < x < 2 and 1 ≤ y1 2 ≤ x and 1 ≤ y 1 (b) When coded messages are sent, there are sometimes errors in transmission. In particular, Morse code uses \dots" and \dashes", which are known to occur in the proportion of 3:4. This means that for any given symbol, P(dot set) = 3 7 and P(dash sent) = 4 7 : Suppose there is interference on the transmission line, and with probability 18 a dot is mistakenly received as a dash, and vice versa. If we receive a dot, what is the probability that a dot was sent? That is, compute P(dot sentjdot received). [7 pts] Answer: Using Bayes’ Rule, we write P(dot sentjdot received) = P(dot receivedjdot sent)P(dot sent)=P(dot received) = (7=8) × (3=7) (7=8) × (3=7) + (1=8) × (4=7) = 21 25 : 2 2 Maximum Likelihood [12 pts] (a) The independent random variables X1; X2; :::; Xn have the common distribution P(Xi ≤ xjα; β) = 8><>: 0; x < 0 (βx)α; 0 ≤ x ≤ β 1; x > β where the parameters α and β are positive. Find the MLEs of α and β. [7 pts] After differentiating, we can get the P.D.F. as P(Xi = xjα; β) = (0βα; otherwise α x(α-1); 0 ≤ x ≤ β Let x(n) be the maximum value. For any fixed α, the likelihood, L(α; βjx) = 0 if β < x(n), and L(α; βjx) is a decreasing function of β if β ≥ x(n). Thus x(n) is the MLE of β For the MLE of α calculate @ @αhnlogα - nαlogβ + (α - 1)log Qi xii = 0 From this we get, α = n nlogx(n) - log Qi xi 3 (b) Suppose that a particular gene occurs as one of the two alleles (A and a), where allele A has frequency θ in the population. That is a random copy of the gene is A with probability θ and a with probability 1 - θ. Since a diploid genotype consists of two genes, the probability of each genotype is given by: genotype | AA | Aa | aa probability | θ2 | 2θ(1 - θ) | (1 - θ)2Suppose we test a random sample of people and we find that k1 are AA, k2 are Aa, and k3 are aa. Find the MLE of θ [5 pts] likelihood = (θ)2k1(2θ(1 - θ))k2(1 - θ)2k3 log likelihood = constant +2k1ln(θ) + k2ln(θ) + k2ln(1 - θ) + 2k3ln(1 - θ) MLE of θ = | 2k1+k2 2k1+2k2+2k3 [Show More]

Last updated: 1 year ago

Preview 1 out of 12 pages

Add to cart

Instant download

We Accept:

We Accept
document-preview

Buy this document to get the full access instantly

Instant Download Access after purchase

Add to cart

Instant download

We Accept:

We Accept

Reviews( 0 )

$9.00

Add to cart

We Accept:

We Accept

Instant download

Can't find what you want? Try our AI powered Search

OR

REQUEST DOCUMENT
56
0

Document information


Connected school, study & course


About the document


Uploaded On

May 13, 2022

Number of pages

12

Written in

Seller


seller-icon
Kirsch

Member since 4 years

907 Documents Sold


Additional information

This document has been written for:

Uploaded

May 13, 2022

Downloads

 0

Views

 56

Document Keyword Tags

Recommended For You

Get more on EXAM REVIEW »
What is Browsegrades

In Browsegrades, a student can earn by offering help to other student. Students can help other students with materials by upploading their notes and earn money.

We are here to help

We're available through e-mail, Twitter, Facebook, and live chat.
 FAQ
 Questions? Leave a message!

Follow us on
 Twitter

Copyright © Browsegrades · High quality services·