Stat 1793

Stat 2593 Chapter 4 Continuous Random Variables and Probability Distributions Here we will look at the probability distribution (some texts use the term
densities instead of distributions for the continuous case) for certain
continuous random variables. There are many types of continuous
probability distributions: exponential, Cauchy, gamma, Weibull...etc, but
the two we will look at are the Continuous uniform distribution and the
normal distribution. We have already looked at the discrete case of the
uniform distribution, that of the number of heads in a toss of a coin or
the number of dots in the toss of one die. Requirements for a Continuous Probability Density Function f(x) For a continuous random variable X with pdf f(X) 1. Total area under the curve must equal one. [pic]
2. [pic] Note: For a continuous random variable P(X=x)=0 Note: [pic]= [pic] Continuous Uniform Probability Density If X is uniform on the interval [pic] then the probability density function
of X is
[pic] with [pic] and [pic]. Example:
A random variable X is uniformly distributed on the interval [pic]. Find
a)the probability density function
b) the mean and standard deviation of X
c) P(4 The normal distribution is a continuous, symmetric, bell-shaped
distribution of a random variable.
> This distribution is also known as the bell curve or the Gaussian
distribution, named for the German mathematician Carl Friederich Gauss
(1777 - 1855) who derived its equation.
> The equation for the normal probability density for a random variable
X with mean ( and standard deviation ( is as follows: [pic], where x(R
and e=2.718...., and ?= 3.14..... This looks scary, but we will be
using tables instead of this formula.
> The area under the normal curve is more important than the
frequencies.
> The shape and position of the normal probability curve is based on the
mean and the standard deviation.
> The curve never touches the x axis but it gets increasingly close. Standard Normal Probability Distribution > The standard normal probability distribution is a normal distribution
with ( = 0 and ( = 1
> The equation for the standard normal probability distribution is
[pic]. Fortunately, tables for the areas or probabilities associated
with the standard normal distribution can be found in Appendix A pages
A-6, 7, Table A.3
> All normally distributed random variables can be transformed into the
standard normally distributed variable by using the formula for the
standard score [pic] or [pic] (we will be using this in the next
chapters) Examples: Find
a) P(Z a +0.5) |
|4. [pic] |P( X < a + 0.5) |
|5. P( X < a) |P( X < a - 0.5) |
Example: A magazine reported that 6 % of North American drivers used the cell phone
while driving. If 300 drivers are selected at random, find the probability
that exactly 25 say they use the cell phone while driving.
0.0227 Example: Of the members of a bowling league, 10 % are widowed. If 200 bowling league
members are selected at random, find the probability that at least 10 are
widowed. 0.9934 Example: If a baseball player's batting average is 0.320 (32 %), find the
probability that the player will get at most 26 hits in 100 times at bat.
0.1190 Section 4.3 exercises pages 162 to 165