Georgia Tech
Fall 2012
Heinrich Matzinger
Practice for nal for math3215
0) This is to help prepare for the nal, a so called practise exam with solutions. On the nal
I will put for sure also linear reg
MATH 3215 Houdré
Test II \
‘ Name:
Answer all questions; write neatly, show all work for full credit; closed books, no calcula-
tors. The point value for each problem is indicated. THE HONOR CODE APPL
1. For a particular instant lottery, the gambler can determine immediately whether or not a purchased
ticked is a winning ticket. Suppose that the probability of a winning ticket is p = 1/10.
(10pts)
Math 3215: Final Exam Sample Questions
Will Perkins
April 26, 2012
1
Basic Probability
1. Write a formula for Pr[A B C ] in terms of probabilities of intersections of events.
2. What is the dierence b
MATH 3215 Houdré
FINAL EXAM E
I ' " Name: M f
Answer all questions; show all work; closed books, no calculators. Tables are included.
THE HONOR CODE APPLIES
‘ Problem Points Score ‘ 1. A, B and C a
2
2. Suppose that the diameter at breast height of a certain type of tree is normally distributed with = 8.8
and = 2.8 (in inches).
(18 points)
(a) What is the probability that the diameter of a rando
Georgia Tech
Fall 2012
Heinrich Matzinger
Solution set for homework 8 for math3215
Name:
1) We throw a fair six-sided die twice. What is the probability that the sum of the numbers is
equal to 8? We a
4.4. Continuous bivariate distributions
Math 3215
Fall 2017
Observation: We will define bivariate distributions for continuous random variables
Definition 1. If X and Y are two continuous random varia
5.6. The central limit theorem
Math 3215
Fall 2017
of a random sample of size n from a distribution with
Observation: As we saw before, the mean X
remains
= and 2 = 2 . So as n increases, the mean
6.1. Descriptive statistics
Math 3215
Fall 2017
Observation: When recording instances of a continuous random variable (mass, concentration,
salary, etc), the data often looks like discrete data and we
4.1 Discrete Bivariate distribution
Math 3215
Fall 2017
Observation: So far we have only seen distributions associated to one random variable. In Chapter
4 we extend the concept to multiple variables
5.8.Chebyshevs inequality
Math 3215
Fall 2017
of a random sample of size n from a distribution with mean
Last time: We saw that the mean X
converges to in some sense.
is N (, 2 /n), so as n increas
7.4 Sample size
Math 3215
Fall 2017
Question: We know that the confidence interval gets smaller the larger the sample size. How big
does the sample have to be to ensure a certain confidence interval?
7.1 Confidence intervals
Math 3215
Fall 2017
Last time: We proved that for a random sample of size n out of the normal distribution N (, 2 ),
is an unbiased estimator for (and the varianve of the emp
Outline
Weak Convergence of Random Variables
The Central Limit Theorem
Central Limit Theorem
Math 3215: Fall 2016
November 10, 2016
Math 3215: Fall 2016
Central Limit Theorem
Outline
Weak Convergence
4.2. The Correlation coefficient
Math 3215
Fall 2017
Definition 1. If u(X, Y ) = (X X )(Y Y ), then the expectation E(X X )(Y Y ), denoted
XY or Cov(X, Y ), is called the covariance of X and Y
Intuiti
Math 3215 Sample Final Exam
Fall 2016
SOLUTlGNS
THIS IS A PRACTICE EXAM AND SHOULD BE USED FOR PRACTICE ONLY. IN ORDER TO
TAKE INTO ACCOUNT THE LENGTH OF THE EXAM, SOME TOPICS THAT ARE INCLUDED
IN THE
Review 3, Math 3215, Spring 2016
Problem 1. Let X1 , X2 be a random sample of size n = 2 from a distribution with pdf
f (x) = 3x2 , 0 < x < 1. Determine
(a) P (max Xi < 3/4) = P (X1 < 3/4, X2 < 3/4).
Review 1, Math 3215, Spring 2016
Problem 1. In the gambling game craps, a pair of dice is rolled and the outcome of the
experiment is the sum of the points on the up sides of the six-sided dice. The b
7.3 Confidence intervals for proportions
Math 3215
Fall 2017
Question: We said that, when drawing histograms, the relative frequencies approximate the probability of the given interval. But how close
3.3 The normal distribution
Math 3215
Fall 2017
Definition 1. A random variable whose probability density function is given by
(x)2
1
f (x) = e 22
2
for all < x < and for parameters < < and 0 < < is
4.3. Conditional distributions
Math 3215
Fall 2017
Observation: Let X, Y be two discrete random variables with joint pmf given by f (x, y). Let A be
the event that X = x and let B be the event that Y
3.2 Exponential, Gamma, Chi Square
distributions
Math 3215
Fall 2017
The exponential distribution
Example 1. On average, your website receives 3 hits a day. What is the probability of receiving no hit
5.5. Random functions associated with the
normal distribution
Math 3215
Fall 2017
Recall: If X1 , X2 .Xn are random variables with means 1 , 2 .n and variances 12 .n2 , then
P
P
the mean of Y = ni=1 c
5.3. Several independent random variables
Math 3215
Fall 2017
Definition 1. If X1 , X2 , .Xn are mutually independent random variables with the same pdf (or pmf)
f (x), then the collection X1 , X2 , .
5.1. Functions of a random variable
Math 3215
Fall 2017
Goal: Given a random variable X, and its pdf, can we find the cummulative distribution and pdf of
u(X), where u is some function of X, i.e. can
5.9. Limiting moment generating functions
Math 3215
Fall 2017
Goal: To prove the central limit theorem.
Theorem 1. Let Mn (t) be a sequence of moment generating functions and suppose limn Mn (t) =
M (
6.4 Maximum likelihood estimators
Math 3215
Fall 2017
Goal: To draw conclusions about the closeness of an estimate to an unknown parameter
Idea: Suppose we know the pdf of a random variable depends on
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Math 3215 (Fall 2017) Final Exam - Page 2 of 13 December 8, 2017
1. (10 points) Determine if the following statements are true (T)
Midterm 1
Math 3215 (Fall 2017)
Introduction to probability and statistics
September 29, 2017
Time Limit: 50 Minutes
Name:
ID:
This exam contains 6 pages (including this cover page) and 5 problems. Ch
1
xa
F (y) = 1
k=0
P1
Two sided confidence interval for proportions:
or for unknown
k
ey (y)
k!
x,
a<b<x
b<x
Two sided confidence intervals for means:
Normal
Chi-square
Gamma
Exponential
F (x) =
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