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Statistics notes Andre Sostar November 27, 2014
1 Discrete Random Variables
5 Bernoulli Random Variables
6 The Binomial Distribution
7 The Geometric Distribution
8 The Negative Binomial Distribution
8 The Hypergeometric Distribution
9 The Poisson Distribution
9 2 Continuous Random Variables
10 The Exponential Density
11 The Gamma Density
11 The Beta Density
12 The Normal Distribution
12 3 Functions of a Random Variable
13 The cdf Method
13 The Transformation Method
Generating pseudorandom variables 4 The Expected Value
The Expected Value of a random variable
15 Calculating Expected Values
15 Expected values of functions of random variables
15 5 The Variance
16 The Variance of a random variable
16 Chebyshev's inequality
17 6 Moment Generating Functions
18 Moment Generating Functions
18 7 Joint and Marginal Distributions
20 Multivariate Random Variables
20 Continuous r.v. joint pdf
20 Marginal cdf and marginal pdf
21 8 Independent Random Variables
23 Independent Random Variables
23 Conditional probability mass function
23 Conditional probability density function
24 The Law of Total Probability
24 9 Functions of Jointly Distributed r.v's
25 The Method of Distribution Functions
25 The Distribution of the Sum
The Method of Moment Generating Functions
25 The Moment Generating Functions for sums and means of i.i.d. r.v. 25 Expected values functions of jointly distributed r.v. 10 Covariance and Correlation
The concept of dependence
27 Linear dependency
27 The Correlation Coecient
28 Covariance of linear combinations
28 11 Conditional expectations
30 Conditional expectations
30 Conditional variance
30 12 The Law of Large Numbers
31 The Law of Large Numbers
31 13 Convergence in distribution
32 Convergence in distribution
32 Theorem proving Convergence in distribution
32 14 The Central Limit Theorem
The Central Limit Theorem 15 Other Important Laws
The Multiplicative Law
34 The Multiplicative Law (bivariate)
34 The conditional pmf/pdf
34 Hierarchic Models
34 Independent Addition Law
Discrete Random Variables
Denition: The probability that Y takes on the value y,
P (Y = y), is
dened as the sum of the probabilities of all sample points in S that are assigned the value y. We will sometimes denote P (Y = y) by p(y). For any discrete probability distribution, the following must be true:
1. 0 [?] p(y) [?] 1 for all y.
p(y) = 1, where the summation is over all values of y with nonzero probability.
Denition: A random variable on is a real valued function X : - R, that is a function that assigns a real value to each possible outcome in .
Denition: A discrete random variable is a random variable that can take on only a nite or at most a countably innite number of values. Denition: The probability mass function (pmf ), is the probability measure on the sample space that determines the probabilities of the various values of X; is those values are denoted by x1 , x2 , * * * , then there is a function p such that p(xi ) = P (X = xi )
p(xi ) = 1.
i How to determine if the discrete random variable is one of the common types.
Question 1. Is it possible to interpret the random experiment as sampling
from an urn with only two types of balls (e.g. successes and failures)?
Question 2. If so, is the draw done with or without replacement, i.e. is the probability of success always the same?
Question 3. Is the number of balls to be drawn xed, or are we to draw balls until some criterion is met?
Question 4. In what way does our random variable evaluate the outcome of the random experiment, i.e. the drawn balls?
5 The cumulative distribution function (cdf ) of a random variable, is dened to be
F (x) = P (X [?] x),
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