A. P. STATISTICS
LESSON 7 – 1 ( DAY 1 )
DISCRETE AND CONTINUOUS RANDOM
VARIABLES
ESSENTIAL QUESTION:
What are discrete and continuous
random variables, and how are
they used to determine
probabilities?
• To define discrete random variables.
• To use discrete random variables to solve
probability problems.
• To define continuous random variables.
• To use continuous random variables to solve
probability problems.
Introduction
• Sample spaces need not consist of
numbers. When we toss four coins, we can
record the outcomes as a string of heads
and tails, such as HTTH.
• Random Variable – A random variable is a
variable whose value is a numerical
outcome of a random phenomenon.
Discrete and Continues
Random Variables
Discrete and Continuous are two
different ways of assigning
probabilities.
Discrete Random Variables
A discrete random variable X has a countable number of
possible values. The probability of X lists the values and
their probabilities:
Values of X: x1 , x2 , x3 , …. Xk
Probability: p1 , p2 , p3 , …pk
The probabilities pi must satisfy two requirements:
1. Every probability pi is a number between 0 and 1.
2. p1 + p2 + …. + pk = 1
Find the probability of any event by adding the
probabilities pi of the particular values xi that make up the
event.
Example 7.1
Getting Good Grades
Page 392
• When using a histogram the height of each
bar shows the probability of the outcome at its
base.
• Because the heights are probabilities, they
add up to 1.
• The bars are the same width.
Example 7.2
Tossing Coins
Page 394
Two assumptions are made:
1. The coin is balanced, so each toss is
equally likely to give H or T.
2. The coin has no memory, so tosses are
independent.
Continuous Random Variables
When using the table of random digits to
select a digit between 0 and 9, the result is a
discrete random variable.
The probability model assigns probability
1/10 to each of the 10 possible outcomes.
Continuous random variable is a number at
random from 0 to 1.
Continuous Random Variables
A continuous random variable X takes all values in
an interval of numbers.
The probability distribution of X is described by a
density curve.
The probability of any event is the area under the
density curve and above the values of X that
make up the event.
Discrete and Continuous
Random Variables
Exercises
Page 395 - 397
•
•
•
•
•
7.1 Roll of the Die
7.2 Three Children
7.3 Social Class in England
7.4 Housing in San Jose, I
7.5 Housing in San Jose, II
Example 7.3
Random Numbers and the
Uniform Distribution
Page 398
Continuous Random Variables
We assign probabilities directly to events – as
areas under a density curve. Any density curve
has area exactly 1 underneath it, corresponding to
total probability 1.
All continuous probability distributions assign
probability of 0 to every individual outcome.
We can see why an outcome exactly to .8 should
have probability of 0.
Normal Distributions as
Probability Distributions
The density curves that are most familiar to
us is the normal curves.
Normal distributions are probability
distributions. Recall that N(μ, σ) is our
shorthand notation for the normal
distribution having mean μ and standard
deviation σ.
Z=X-μ
σ