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
- 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
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.