Lecture 11 Notes, Nonparametric Statistics
Does not depend on the population fitting any particular type of distribution
(e.g, normal). Make fewer assumptions and apply more broadly at the
expense of a less powerful test (needing more observations to draw
Lecture 10 Notes, Hypothesis Testing
1 Definitions
1.1
Hypothesis Testing
A (parametric) hypothesis is a statement about one or more population
parameters. This hypothesis can be tested using a hypothesis test.
A hypothesis test consists of:
1. Two comple
Lecture 9 Notes, Intervals
1 Interval Estimation
Interval estimation is another approach for estimating a parameter . Interval estimation
consists in finding a random interval that contains the true parameter with probability (1
). Such an interval is ca
Lecture 8 Notes, Estimation
1 Definitions
1.1
Parameter
A pmf/pdf can be equivalently written as fX (x) or fX (x|), where represents the
constants that fully define the distribution. For example, if X is a normally distributed RV,
the constants and will f
Lecture 2 Notes, Discrete and Random Variables
Random Variables
1
1.1
Intuitive Definition
A random variable is a variable with unknown numerical value that can take on, or
represent, any possible element from a sample space. The elements of a sample
spac
Lecture 6 Notes, Normal Distribution
1 Discrete Distributions
1.1
Hypergeometric Distribution
Let the RV X be the total number of successes in a sample of n elements drawn
from a population of N elements with a total number of M successes. Then, the
pmf o
Lecture 7 Notes, Random Sampling
1 Definitions
1.1
Random Sample
Let X1., Xn be mutually independent RVs such that fXi (x) = fXj (x) i = j. Denote fXi (x) = f(x). Then, the collection
X1, ., Xn is called a random sample of size n from the population f(x).
Lecture 5 Notes, Vectors
1 Function of a Random Variable (Univariate Model)
1.1
Discrete Model
Let X be a discrete random variable with pmf fX (x). Define a new random variable Y
as a function of X, Y = r(X). The pmf of Y , fY (y), is derived as follows:
Lecture 1 Notes, Set and Probability
1
Set Theory
1.1
Definitions and Theorems
1. Experiment: any action or process whose outcome is subject to uncertainty.
2. Sample Space: collection of all possible outcomes (or elements) of the
experiment (set S). [Fin
Lecture 4 Notes, Expectations
1 Expected Value
1.1 Univariate Model
Let X be a RV with pmf/pdf f (x). The expected or mean value of X, denoted E(X)
or X , is defined as:
E(X) = X =
xf (x)
(discrete model)
x X
(1)
E(X) = X =
xf (x)dx
(continuous model)
In
Lecture 3 Notes, Multivariable Distribution
1 Multiple Random Variables
1.1
Bivariate Distribution
Many experiments deal with more than one source of uncertainty. For these cases a random
vector must be defined to contain the multiple random variables we
Lecture 10 Notes, Regression
Regression analysis allows us to estimate the relationship of a response
variable to a set of predictor variables
Let
x1, x2, xn
be settings of x chosen by the investigator and
y1, y2, yn
be the corresponding values of the res
Lecture 2 Notes, Data
A population is a collection of objects, items, humans/animals (units) about
which information is sought.
A sample is a part of the population that is observed.
A parameter is a numerical characteristic
Lecture 9 Notes, Two-Sample Inference
Independent Samples Design:
There are a few dierent ways we can do an experiment. In an independent samples design,
we have an independent sample from each population. The data from the two groups are independent.
Sa
Lecture 6 Notes, Inference
Statistical Inference is the process of making conclusions using data that is subject to random variation.
Bias() := E() , where is the true parameter value and is an estimate of it
computed from data.
Mean-Squared Error (MSE)
Lecture 4 Notes, Central Limit
Let X1, X2, . . . , Xn be a random sample drawn from any distribution with a finite mean and
2
variance . As n , the distribution of:
X
/
n
converges to the distribution N(0, 1). In other words,
X
N(0, 1).
/ n
Note 1: What
Lecture 5 Notes, Confidence Intevals
Instead of reporting a point estimator, that is, a single value, we want to report a
confidence interval [L, U] where:
P cfw_L U = 1 ,
the probability of the true value being within [L, U] is pretty large.
Here, [L, U]
Lecture 1 Notes, Probability
A probability space, defined by Kolmogorov (1903-1987) consists of:
A set of outcomes S, e.g.,
for the roll of a die, S = cfw_1, 2, 3, 4, 5, 6,
for the roll of two dice, S =
1 , 1 , 2 , 1 ,., 6
1
2
1
3
6
temperature on Monday
Lecture 8 Notes, Single Sample Inference
You know already for a large sample, you can invoke the CLT so:
2
X N(, ).
Also for a large sample, you can replace an unknown by s.
know how to do a hypothesis test for the mean, either:
calculate z-statistic a
Lecture 3 Notes, Numerical Data
Sample Mean:
x
n
i =1 i
x :=
.
n
Sample median: order the data values x(1) x(2) x(n), so then
x( )
n odd
n+1
2
median := x := 1
2
n
n
[x( )
+ x( +1)]
2
n even
.
2
Mean and median can be very dierent: 1, 2, 3, 4, 500 .
o