Trinh Tat
9/20/16
Worksheet 8
Random Variables and Distribution Functions
1. Let X be a discrete random variable with probability mass function
(a) Find c. (all the values of fx(x) = 1
4c + 2c + c + 4c + 2c = 1
13c = 1
c = 1/13
(b) Find Pcfw_X > 0 and Pcf
Dani McEachern
Math 363
April 19, 2016
Worksheet 22
t Procedures
1. A rule of thumb for maximum heart rate is 220age beats per minute where age is measured in
years. Thus, the mean maximum heart rate for 22 year-olds would be 198 beats per second. You
wan
Goodness of Fit/Analysis of Variance
Worksheet 23
1. For a dihybrid cross, if the assortment is independent, we expect the odds of the 4 phenotypes to be
9:3:3:1
for respectively,
both dominant: dominant, recessive: recessive, dominant: both recessive
tra
Dani McEachern
March 24, 2016
Worksheet 16
Method of Moments
1. Daily rainfall data, in millimeters, is modeled as having a (1/2, ) distribution. The density is
0 for x< 0
1 /2
f X (x , )=
x1/ 2ex for x 0
a. Find the method of moments estimator for base
Dani McEachern
Math 363
April 5, 2016
Worksheet 19
Simple Hypotheses
1. The body temperature in degrees Fahrenheit of 52 randomly chosen healthy adults is measured
with the following summary of the data:
n = 52, x = 98.33 = 0.68.
a. Are the necessary cond
Dani McEachern
Math 363
January 21, 2016
Worksheet 2
Describing Distributions with Numbers
1. The 2014 Arizona Diamondbacks had the worst record in Major League Baseball Team. Below is
the run differential for the 162 games.
Run
Differential
# of Games
-1
Dani McEachern
Math 363
February 4, 2016
Worksheet 6: Basics of Probability
1. Roll three fair dice.
a. Keeping track of the order, how many different outcomes are possible?
6*6*6= 216 outcomes possible
b. What values are possible for the sum on the three
Dani McEachern
Math 363
February 23, 2016
Worksheet 11
Examples of Mass Functions and Densities
1. In this problem, we shall use R to calculate probabilities and quantiles for random variables.
a. For X a negative binomial with n = 4 and p = 3/7, find Pcf
Dani McEachern
Math 363
February 16, 2016
Worksheet 9
Random Variables and Distribution Functions
1. Let X be a discrete random variable with probability mass function
x
0
1
x
0 1 2 3
fX(x) 1/6 2/6=1/3
fX(x) c 2c 2c c
2
2/6=1/3
3
1/6
a. Find c.
c=1/6
b. F
Dani McEachern
Math 363
January 28, 2016
Worksheet 4
Correlation and Regression
1. Observations made by the astronomer Edwin Hubble showed that the universe is expanding. If v
is the galaxys recession from the Milky Way and d is the distance to that galax
Dani McEachern
Math 363
April 14, 2016
Worksheet 21
Extensions on the Likelihood Ratio Test
1. One of the lenses in your supply is suspected not to have the focal length f of 9 centimeters as
claimed by the manufacturer.
a. Write an appropriate hypothesis
Dani McEachern
Math 363
March 10, 2016
Worksheet 14
Central Limit Theorem
1.
Snells law tell us how light bends at an interface - the angle of incidence versus the angle of
refraction - based on the ratio of the velocities of light in the two isotropic me
Dani McEachern
Math 363
February 2, 2015
Worksheet 5
Producing Data
1. A health study is being conducted on a population of 30 women and 50 men.
a. Label the women 1 through 30 and the men 31 through 80. Estimate many women
are in a simple random sample o
Dani McEachern
Math 363
April 12, 2016
Worksheet 20
Composite Hypotheses
1. Current norms at the blood bank assume a given fraction of individuals have a given blood type.
a. Write a hypothesis to test if more than 20% of the Tucson population has blood t
Dani McEachern
Math 363
February 18, 2016
Worksheet 10
The Expected Value
1. Let X be a discrete random variable whose survival function is shown.
a.
Find
Pcfw_X
= 3.
Pcfw_X=3 = 0.9-0.6=0.3
b. Give the probability mass function for X.
X
0 1
2 3
4
5
fX(x 0
Dani McEachern
Math 363
March 8, 2016
Worksheet 13
Central Limit Theorem
1. Let X be the value on an unfair die with mass functions.
x
1
2
3
4
5
1/12
1/12
1/12
f (x)
x
6
1/4
a. Find the mean and standard deviation for X.
Mean= EX= 1(1/12)+2(1/4)+3(1/12)+4
Dani McEachern
Math 363
February 11, 2016
Worksheet 8
Random Variables and Distribution Functions
1. Let X be a discrete random variable with probability mass function(all values add to
one):
x
-2 -1 0 1 2
fX(x) 3c 2c c 2c 3c
a. Find c.
c = 1/11
b. Find P
Trinh Tat
9-27-16
Worksheet 10
MATH363
The Expected Value
1. Let X and Y be discrete random variables whose survival functions are shown. (X in red and Y in
blue)
(a) Find Pcfw_X = 3
Pcfw_X = 3 = 0.1
(b) Find Pcfw_Y > 1
Pcfw_Y > 1 = 1- Pcfw_Y 1 = 1 0.3 =
Dani McEachern
Math 363
February 9, 2016
Worksheet 7
Conditional Probability and Independence
1. Twins make up a small fraction of the human population. Identical (monozygotic) twins make up
8% of all twins. Identical twins always have the same sex. The o
Dani McEachern
Math 363
March 29, 2016
Worksheet 17
Maximum Likelihood Estimation
1. Daily rainfall data, in millimeters, is modeled as having a (1/2, ) distribution. The density is
0 for x< 0
f X (x , )= 1 /2 1/ 2 x
x e for x 0
cfw_
a. Find the maximum
Dani McEachern
Math 363
March 22, 2016
Worksheet 15
Method of Moments
1. For a parameter > 0, we model the accuracy of a dart player by the density
0 if x< 0
f X ( x| )= x 1 if 0 x <1
0 if 1 x
for a continuous random variable X, the distance the dart is f
Dani McEachern
Math 363
March 3, 2016
Worksheet 12
Law of Large Numbers
1. Consider the function g(x) = 3*arctan(x)/(x4 + 1)
a. Plot g on the interval from 0 to 3.
> y<-function(x) 3*atan(x)/(x^4)+1)
> curve(y,0,3)
3
b. Estimate the integral
g ( x ) dx
u
Dani McEachern
Math 363
March 31, 2016
Worksheet 18
Maximum Likelihood Estimation
1. Loss of property for insurance purposes is sometimes modeled as a Pareto distribution. If we take
the loss (in thousands of dollars), this yields a density of
5
f ( x| )
Dani McEachern
Math 363
January 26, 2016
Correlation and Regression Worksheet 3
1. Global warming has many indirect effects on climate. For example, summer monsoon winds in the
Arabian Sea bring rain to India needed for agriculture. As the climate warms a