Random Variables

1
Random Variables

• Discrete Bernoulli, Binomial, Geometric,
  Poisson
• Continuous Uniform, Exponential, Gamma, Normal
• Expectation Variance, Joint Distributions,
  Moment Generating Functions, Limit Theorems

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.

3
pmfs and cdfs

• 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,

4
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) PX 0 1 p and p(1) PX 1
  p
• X is a Bernoulli random variable with parameter p.

5
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.

6
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

7
Poisson Random Variable

• X is a Poisson random variable with parameter l gt
  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.

8
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.
9
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
10
Exponential random variable

• X has an exponential distribution with parameter
  l gt 0 if its pdf is
• Then its cdf is
• This distribution has very special
  characteristics that we will use often!

11
Gamma random variable

• X has an gamma distribution with parameters l gt 0
  and a gt 0 if its pdf is
• It gets its name from the gamma function
• If a is an integer, then

12
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 aXb is
  normally distributed with parameters amb and
  (as)2 . In particular, Z (X m)/s has the
  standard normal distribution with parameters 0
  and 1.

13
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.

14
Expectations of Discrete Random Variables

• Bernoulli EX 1(p) 0(1-p) p
• Binomial EX np
• Geometric EX 1/p (by a trick, see text)
• Poisson EX 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

15
Expectations of Continuous Random Variables

• Uniform EX (a b)/2
• Exponential EX 1/l
• Gamma EX ab
• Normal EX m the first parameter is the
  expected value note that its density is
  symmetric about x m

16
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

17
Higher-order moments

• The nth moment of X is EXn
• The variance is
• It is sometimes easier to calculate as

18
Variances of Discrete Random Variables

• Bernoulli EX2 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
  EX)
• Poisson Var(X) l the parameter is also the
  variance of the number of rare events per
  interval!

19
Variances of Continuous Random Variables

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

20
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

21
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,

22
Covariance

• The covariance of X and Y is
• If X and Y are independent then Cov(X,Y) 0.
• Properties

23
Variance of a sum of r.v.s

• If X1, X2, , Xn are independent, then

24
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