Class Notes, Class 7

Chapter 6 - Continuous Random Variables

f(x) is probability density

p(x) = 0 for any particular value of x

integrate over intervals to get probabilities

F(x) will be useful - changes integrals to subtractions - can do 3 ways:

1. do integrals every time

2. get a formula for F(x) by doing the integral once

3. use tables that are pre-calculated

Continuous Uniform Distribution

The density function for a continuous uniform distribution over the interval [A, B]:

Derivations of the mean, variance, and cumulative distribution:

Normal Distribution - the workhorse of distributions

The density function for a normal distribution:

The mean, variance, and cumulative distribution:

Special properties of the normal distribution

it models many real-world processes

it is a good approximation to many other distributions, for example the binomial and the Poisson

the parameters are just mu, sigma

the mean and variance are independent quantities; this is unique

it is the limiting distribution for the sampling distribution of means for any distribution - Central Limit Theorem

for a bivariate normal distribution, COV(X, Y)=0 -> X, Y marginals independent

the sum of normal random variables is itself normal

Standard Normal Distribution

The density function for a standard normal distribution

The mean, variance, and cumulative standard normal distribution:

Use of tables, both forwards and backwards

Approximations of discrete distributions

Poisson approximates binomial when p near 0 or near 1, and n large (work using the smaller of p or q). The binomial tables in the back of the book work up to n = 20. Poisson distribution tabled up to lambda = 18.8

Use

Normal approximates binomial when p near 1/2, and n large enough; again, binomial tables in the back of the book work up to n = 20. If p near 0 or 1, but n very, very large so that the Poisson approximation is out of range (np > 18.8), the normal approximation can be used in that situation as well.

Use and convert to Z.

Normal approximates Poisson when , although your tables are good up to lambda = 18.8.

Use and convert to Z.

Correction for Continuity:

The Poisson distribution and the binomial distribution are both discrete random variables, so no correction is required when the one approximates the other.

If the normal distribution is used to approximate either the Poisson or the binomial, so that a continuous variable is being used to approximate a discrete one, then the correction can be used.

The correction simply involves treating each value as a little interval in itself: 16 becomes [15.5 to 16.5], for instance. Use common sense in applying to problems.