If X is exponential with parameter , such that fX (x) = exp(x), show that the ceiling of X (i.e., the next integer, so that the integer of 1.1 is 2, the integer of 2.4 is 3 etc) Y has geometric distribution with parameter p = 1 exp(). Hint: Consider P (Y
Todays class (207-255)
Re-cap change of variables
Expectation of a random variable and properties
Expectation of a function of a r.v.
Variance
Moments and Moment generating functions
Mean and Median
Covariance and correlation
Conditional expectation
SKIP
Chapter 3 sections
We will SKIP a number of sections
Random variables and discrete distributions
Continuous distributions
The cumulative distribution function
Bivariate distributions
Marginal distributions
Conditional distributions
SKIP Multivariate distr
Covariance and Correlation
Cov (X , Y ) = E (X x )(Y Y ) = E (XY ) X Y .
(X ,Y
XY = CovX Y ) . eg: f (x, y ) = 1, 0 < x < 1, x < y < x + 1,
= 1/ 2.
If X , Y independent, Cov (X , Y ) = X ,Y = 0.
Var (aX + bY + c) = a2 VarX + b 2 Var (Y ) + 2abCov (X , Y
Chapter 3 sections
We will SKIP a number of sections
Random variables and discrete distributions
Continuous distributions
The cumulative distribution function
Bivariate distributions
Marginal distributions
Conditional distributions
SKIP Multivariate distr
Todays class (275-356)
Special distributions
Bernoulli and Binomial
SKIP Hypergeometric
Poisson
SKIM Negative Binomial and Geometric
Normal distribution
Gamma and exponential
SKIM Beta
SKIM Multinomial
Bivariate Normal
Discrete uniform distribution
A disc
1) For the following question we will be using the data points:
X
3
2
5
0
Y
10
8
13
2
a) Draw a Scatter plot for the data points.
b) Find x, SDx , and SDy .
y
2
c) Find SSX, SSY, SXY by using the following table:
X
3
2
5
0
Y
10
8
13
2
(xi x)
(yi y )
(xi x
1. Suppose that a box contains one blue card and four red cards, which are
labeled A, B, C, D. Suppose also that two of these ve cards are selected at
random, without replacement.
(a) If it is known that card A has been selected, what is the probability
t
Linear regression
Have xi s and yi s which are assumed to be linked through
yi = + xi +
How do we nd and ?
i
Least squares
Find and such that they minimize
(yi yi )2 .
We get
=
(xi )(yi )
x
y
(xi )2
x
= y
x
MLE
Now assume in addition that i N(0, 2 ). T
Dimension of hypotheses
Example of nested hypotheses: Xi N(, 2 )
H0 : = 0
H1 : = 0
Under H0 , there is ONE parameter we dont know, 2 (and we
take MLE). So |0 | = 1.
Under H1 , there are TWO parameters we dont know, and 2
(and we take MLE of both). So |1 |
one-sided vs two-sided alternatives
Two-sided:
H0 : Xi N(0 , 2 ), vs
H1 : Xi N(, 2 ), = 0 .
One-sided:
H0 : Xi N(0 , 2 ), vs
H1 : Xi N(, 2 ), > 0 .
Unbiased tests and power functions
H0 : 0 , vs
H1 : 1 .
A test procedure is unbiased if for EVERY 0 and 1 ,
Composite Hypotheses
Example 1
H0 : Xi N(0 , 2 )
H1 : Xi N(1 , 2 ),
2 unknown.
Example 2
H0 : Xi Bin(n = 100, p = 0.5)
H1 : Xi Bin(n = 100, p < 0.5)
Two-sided (or two-tailed) p = 0.5 versus one-sided p < 0.5.
Usually NO TEST will be UNIFORMLY most powerf
Hypothesis testing
Suppose we observe X1 , . . . , Xn and we know that they are iid but
we arent sure what distribution they come from for example, we
might know that exactly one of these two hypotheses is true:
H0 : Xi
Po( = 2)
H1 : Xi
Po( = 3)
How cou
The t-distribution
If X1 , ., Xn N(, 2 ), we know that
X
N(0, 1).
/ n
But what if we dont know 2 ?
1
2
Estimate 1 = n1 n (Xi X )2 , unbiased estimator of 2 .
i=1
Then
X
tn1 .
1 / n
The higher n, the close this distribution is to N(0, 1).
The distribut
Normal distribution
Poisson distribution
Binomial distribution
Gamma distribution
Properties reminder
Var (X + Y ) = Var (X ) + Var (Y ) + 2Cov (X , Y )
Var (X ) = E (Var (X |Y ) + Var (E (X |Y )
E (X ) = E (E (X |Y )
Properties of distributions
If X Bin(
If X is exponential with parameter , show that the ceiling of X (i.e., the next integer) Y has geometric distribution with parameter p = 1 exp(). Hint: Consider P (Y = 1) = P (0 < X < 1), generalize for any i, and compare with the pmf of the geometric wh
1-4. Casella & Berger 2.34, 2.36, 3.4, 3.6 5. Show that if X has Poisson distribution with mean , and Y has Poisson distribution with mean , then X + Y also has Poisson distribution with mean + . Hint: use MGF. 6. Show that the MGF of the negative binomia
1. Construct a rectangle with its two side lengths generated independently from the U nif orm(0, 1) distribution. (a) Find the pdf of the area of this rectangle. (b) Find the pdf of the perimeter of this rectangle. (c) Find the pdf of the area of the larg
HW9
1. Let Xn Beta(n, n), n = 1, 2, 3, . Show that Xn (1/2) [i.e., show that P (|Xn 1/2| > ) 0 as n for every > 0. Use Chebyshevss inequality.] 2. Slutskys theorem states the following: if Xn X and Yn a where D D a = 0 is a constant, then Xn /Yn X/a (here
STA213: Introduction to Statistical Methods
Instructor: Ioanna Manolopoulou
TuThu 2.50-4.05 Physics 130
September 1, 2010
Recap
Venn diagrams and denitions: union, intersection, subset, mutually exclusive Properties of set operations Denition of a probabi
Recap Venn diagrams and denitions: union, intersection, subset, mutually exclusive Properties of set operations Denition of a probability function Example: fair coin, dart board Some theorems and inequalities Todays outline Leftover from last time: Booles
Todays outline: pp 47-55 Examples of variable transformation One-to-one transformations Theorems Example 1: Exponential distn [similar to HW 1.53, 1.55] We are given a random variable X with pdf fX (x) = exp(x), x > 0.
We say that X Exp() is exponentiall
Todays outline: pp 55-68 Re-cap from last class More examples of transformations Expected values Moments and Moment Generating Functions Example 2.3.10 and Thms 2.3.11 and 2.3.12 will be skipped (the only thing you need to know is that existence of MGFs i
Todays outline: pp 98-111
Examples from HW/book Continuous distributions: Uniform Gamma (and exponential and chi square) Normal (Gaussian) Beta Cauchy Lognormal Double exponential Today we will nish the material for the midterm :-) Next class (Thursday) w
Todays outline: pp 226-231 Order statistics 1. Denition 5.4.1: The order statistics of a random sample X1 , ., Xn are the sample values placed in ascending order, denoted X(1) , ., X(n) . 2. Range, Median, upper/lower quartile, interquartile range. 3. Den