Mathematics Notes > Probability and Statistics Notes

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Statistics notes Andre Sostar November 27, 2014

Contents

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

13 1

Generating pseudorandom variables 4 The Expected Value

13 15

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 Moments

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

25 2

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

26 27

The concept of dependence

27 Linear dependency

27 Covariance

27 Covariance

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 Unbiasedness

31 Consistency

31 The Law of Large Numbers

31 13 Convergence in distribution

32 Convergence in distribution

32 Theorem proving Convergence in distribution

32 Standardizing

32 14 The Central Limit Theorem

33 3

The Central Limit Theorem 15 Other Important Laws

33 34

The Multiplicative Law

34 The Multiplicative Law (bivariate)

34 The conditional pmf/pdf

34 Hierarchic Models

34 Independent Addition Law

34 4

1

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.

2. [?]

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 )

and

[?]

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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