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### Random Variables

Discrete: Bernoulli, Binomial, Geometric, Poisson

Continuous: Uniform, Exponential, Gamma, Normal

Expectation & Variance, Joint Distributions, Moment Generating Functions, Limit Theorems

Chapter 2

Definition of random variable

A random variable is a function that assigns a number to each outcome in a sample space.

- If the set of all possible values of a random variable X is countable, then X is discrete. The distribution of X is described by a probability mass function:
- Otherwise, X is a continuous random variable if there is a nonnegative function f(x), defined for all real numbers x, such that for any set B,

f(x) is called the probability density function of X.

Chapter 2

pmf’s and cdf’s

- The probability mass function (pmf) for a discrete random variable is positive for at most a countable number of values of X: x1, x2, …, and
- The cumulative distribution function (cdf) for any random variable X is

F(x) is a nondecreasing function with

- For a discrete random variable X,

Chapter 2

Bernoulli Random Variable

- An experiment has two possible outcomes, called “success” and “failure”: sometimes called a Bernoulli trial
- The probability of success is p
- X = 1 if success occurs, X = 0 if failure occurs

Then p(0) = P{X = 0} = 1 – p and p(1) = P{X = 1} = p

X is a Bernoulli random variable with parameter p.

Chapter 2

Binomial Random Variable

- A sequence of n independent Bernoulli trials are performed, where the probability of success on each trial is p
- X is the number of successes

Then for i = 0, 1, …, n,

where

X is a binomial random variable with parameters n and p.

Chapter 2

Geometric Random Variable

- A sequence of independent Bernoulli trials is performed with p = P(success)
- X is the number of trials until (including) the first success.

Then X may equal 1, 2, … and

X is named after the geometric series:

Use this to verify that

Chapter 2

Poisson Random Variable

X is a Poisson random variable with parameter l > 0 if

note:

X can represent the number of “rare events” that occur during an interval of specified length

A Poisson random variable can also approximate a binomial random variable with large n and small p if l = np: split the interval into n subintervals, and label the occurrence of an event during a subinterval as “success”.

Chapter 2

Continuous random variables

A probability density function (pdf) must satisfy:

The cdf is:

means that f(a) measures how

likely X is to be near a.

Chapter 2

Uniform random variable

X is uniformly distributed over an interval (a, b) if its pdf is

Then its cdf is:

all we know about

X is that it takes a

value between a and b

Chapter 2

Exponential random variable

X has an exponential distribution with parameter l > 0 if its pdf is

Then its cdf is:

This distribution has very special characteristics that we will use often!

Chapter 2

Gamma random variable

X has an gamma distribution with parameters l > 0 and a > 0 if its pdf is

It gets its name from the gamma function

If a is an integer, then

Chapter 2

Normal random variable

X has a normal distribution with parameters m and s2 if its pdf is

This is the classic “bell-shaped” distribution widely used in statistics. It has the useful characteristic that a linear function Y = aX+b is normally distributed with parameters am+b and (as)2 . In particular, Z = (X – m)/s has the standard normal distribution with parameters 0 and 1.

Chapter 2

Expectation

Expected value (mean) of a random variable is

Also called first moment – like moment of inertia of the probability distribution

If the experiment is repeated and random variable observed many times, it represents the long run average value of the r.v.

Chapter 2

Expectations of Discrete Random Variables

- Bernoulli: E[X] = 1(p) + 0(1-p) = p
- Binomial: E[X] = np
- Geometric: E[X] = 1/p (by a trick, see text)
- Poisson: E[X] = l : the parameter is the expected or average number of “rare events” per interval; the random variable is the number of events in a particular interval chosen at random

Chapter 2

Expectations of Continuous Random Variables

- Uniform: E[X] = (a + b)/2
- Exponential: E[X] = 1/l
- Gamma: E[X] = ab
- Normal: E[X] = m : the first parameter is the expected value: note that its density is symmetric about x = m:

Chapter 2

Expectation of a function of a r.v.

- First way: If X is a r.v., then Y = g(X) is a r.v.. Find the distribution of Y, then find
- Second way: If X is a random variable, then for any real-valued function g,

If g(X) is a linear function of X:

Chapter 2

Higher-order moments

The nth moment of X is E[Xn]:

The variance is

It is sometimes easier to calculate as

Chapter 2

Variances of Discrete Random Variables

- Bernoulli: E[X2] = 1(p) + 0(1-p) = p; Var(X) = p – p2 = p(1-p)
- Binomial: Var(X) = np(1-p)
- Geometric: Var(X) = 1/p2 (similar trick as for E[X])
- Poisson: Var(X) = l : the parameter is also the variance of the number of “rare events” per interval!

Chapter 2

Variances of Continuous Random Variables

- Uniform: Var(X) = (b - a)2/2
- Exponential: Var(X) = 1/l
- Gamma: Var(X) = ab2
- Normal: Var(X) = s2: the second parameter is the variance

Chapter 2

Jointly Distributed Random Variables

See text pages 46-47 for definitions of joint cdf, pmf, pdf, marginal distributions.

Main results that we will use:

especially useful with indicator r.v.’s: IA = 1 if A occurs, 0 otherwise

Chapter 2

Independent Random Variables

X and Y are independent if

This implies that:

Also, if X and Y are independent, then for any functions h and g,

Chapter 2

Covariance

The covariance of X and Y is:

If X and Y are independent then Cov(X,Y) = 0.

Properties:

Chapter 2

Moment generating function

The moment generating function of a r.v. X is

Its name comes from the fact that

Also, if X and Y are independent, then

And, there is a one-to-one correspondence between the m.g.f. and the distribution function of a r.v. – this helps to identify distributions with the reproductive property

Chapter 2

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