Lecture 25
July 4th 2012
A potter is producing teapots one at a
time. Assume that they are produced
independently of each other and with
probability p the pot will be satisfactory;
the rest are sold at a lower price. The
number, X, of rejects produced be
Lecture 25
July 6th 2012
Example: Rain- No rain
Suppose that the probability that tomorrow
is rainy given that today is not raining is .
And that the probability that tomorrow is
dry (not raining) given that today is rainy is
. If tomorrows weather depen
Lecture 22
June 25th 2012
Recall
Moment Generating Function:
It follows that if two random variables have
the same m.g.f then they have the same
distribution.
Where the m.g.fs must match for all
values of t, not just at a few points.
The m.g.f can al
Lecture 20
June 20th 2012
The Poisson Distribution
Variability:
The expected value is a useful summary
statistic, however it tells us nothing about the
variability within the set of observations.
For example, say the heights of 300 room
doors are meas
Lecture 19
June 18th 2012
Notes:
1.E[g(X)] can be interpreted as the average
value of g(X) when the process where X is
defined is repeated for an infinite number
of times.
2.As we saw in calculating the expected
value of X, the expected value of g(X) may
Lecture 21
June 22nd 2012
Properties of Mean and Variance:
If a and b are constants and Y=aX+b, then:
1.E(Y)=aE(X)+b and
2.Var(Y)=a2Var(X)
Proof:
Problem 7.4.3:
An airline knows that there is a 97% chance a
passenger for a certain flight will show up,
Lecture 18
June 15th 2012
Expected Value and Variance
Chapter 7
Often it is more helpful to provide a person
with summary statistics than giving full
details of every outcome.
Where one can immediately get a sense
for the data just by knowing for exampl
Lecture 17
June 11th 2012
Sometimes we can combine other models
with the Poisson Process:
For Example:
A very large number of ladybugs is released
in a large orchard. They scatter randomly
so that on average a tree has 6 ladybugs
on it. Trees are the sa
Lecture 31
July 18th
Mean:
Variance:
Exponential Distribution:
Physical setup: In a Poisson Process for
events in time let X: length of time we wait
until the first occurrence. Then X has an
exponential distribution.
Example:
If phone calls to a fire
Lecture 27
July 9th 2012
Exercise 8.26: Waterloo in January is
blessed by many things, but not good
weather. There are never two nice days in
a row. If there is a nice day, we are just as
likely to have snow as rain the next day. If
we have snow or rain,
Lecture 23
June 27th 2012
Functions of variables:
Example: Let U=2(Y-X) in our previous
example. We might now be interested in
finding the probability function of U, which
is a function of r.vs X and Y.
y
f(x,y)
0
1
2
3
0
1/12
1/4
1/8
1/120
x
1
1/6
1/4
Lecture 32
July 18th 2012
Variance:
a) The probability that you have to wait
longer than 15mins for a bus
b) The probability you have to wait more
than 15mins longer, having already been
waiting for 6mins.
Part b) illustrates the memoryless
property of
Lecture 34
July 23rd 2012
a)
b)
c)
9.6 Use of the Normal Distribution in
Approximation
Under certain conditions the normal
distribution can be used to approximate
probabilities for linear combinations of
variables having a non-normal distribution.
This
Lecture 35
July 25th 2012
Recall Fire example:
Suppose fires reported to a fire station
satisfy the conditions for a Poisson
process, with a mean of 1 fire every 4
hours. Find the probability the 500th fire of
the year is reported on the 84th day of the
Lecture 36
July 25th 2012
Example: Let p be the proportion of
Canadians who think Canada should
adopt the US dollar.
a) Suppose 400 Canadians are randomly
chosen and asked their opinion. Let X be
the # who say yes. Find the probability
that the proportio
Lecture 28
July 11th 2012
For calculation purposes we use:
Interpretation of Covariance
X=persons height and Y=persons weight,
these two r.vs will have positive
covariance
Cov(X, Y)>0
X=thickness of the attic insulation in a
house and Y=heating cost
Lecture 16
June 8th 2012
Example:
Coliform bacteria occur in a river water with
an average intensity of 1 bacteria per 10
cubic centimeters (cc) of water. Find
a)The probability there are no bacteria in a
20cc sample of water which is tested, and
b)The p
Lecture 15
6th June 2012
Recall:
Poisson distribution from Binomial
X:# of events of some type.
If X~Poi( ) then the p.f. is given by
Notes:
1. The same idea can be applied when p is close to
1. This occurs because we can interchange the
labels success a
Lecture 2
May 4th 2012
Recall:
Card example: Find the Probability the
card is a club.
P( Club is drawn)= 13/52 = 1/4
We can use the term odds to describe
probabilities where,
The odds of an event A occurring is given
by:
P(A)
1-P(A)
So in the card ex
Stat 230
May 2nd 2012
Dina Dawoud
M3 3126
Course Outline
Lectures: MWF 9:30 to 10:20, M3 1006
Tutorials: W 3:30 to 4:20, M3 1006
Course Notes: Probability: Stat 220/230
Notes
Available at Campus Copy, MC 2018
Supplementary Lecture Slides (found on
LE
Statistics 230, Fall 2010
Midterm Test 1 October 14, 2010 Duration: 75 Minutes
Section: 001 - La Croix (12:30-1:20) 004 - La Croix (1:30-2:20) Instructions: 1. Please indicate your section above.
Family Name: Given Name: ID #: Signature:
002 - Moshksar (1
STAT 230 - Assignment 2
Due in class on Friday November 12
1. Suppose X is a random variable with probability function fx kx0. 3 x-1 , x 1, 2, . a Given that fx is a probability function, find k. b Find EX. Hint: Since 1 t t2 t3 tx
x0
1 , 1-t
|t| 1
by t
STAT 230 - Assignment 2 Solutions
1. Suppose X is a random variable with probability function fx kx0. 3 x-1 , x 1, 2, . a Given that fx is a probability function, find k. Since fx is a probability function k x0. 3 x-1 1. Since
x1
1 t t2 t3 tx
x0
1 , 1-t
Stat 230 - Assignment 3
Due in class on Friday, December 3, 2010
1. A continuous random variable X is said to have the Gamma distribution with parameters > 0 and > 0 if x x-1 - x0 e () f (x) = 0 otherwise. Here, () =
0
x-1 e-x dx is the Gamma function in
Stat 230
Assignment 1 - Solutions
The first three questions consider the process of arranging coloured marbles in a row from left to right. Two marbles of the same colour are to be considered indistinguishable when counting arrangements. 1. (a) Suppose th