Transcript Example
Warm Up: 2003 AP FRQ #2
7.1 Discrete and Continuous
Random Variables
We usually denote random variables by capital
letters such as X or Y
When a random variable X describes a random
phenomenon, the sample space S just lists the
possible values of the random variable.
Example: The count of heads in four tosses of a
coin
Let X = count of heads in 4 tosses
X
0
1
2
3
4
P(x)
1/16
4/16
6/16
4/16
1/16
What is the probability
distribution of the discrete
random variable X that counts
the number of heads in four
tosses of a coin?
Probability of tossing at least 2
heads?
Probability of at least one
head?
Example
The instructor of a large class gives 15% each of
A’s and D’s, 30% each of B’s and C’s, and 10%
F’s. Choose a student at random from this class.
The student’s grade on a 4-pt scale (A = 4) is a
random variable X. Find the probability that the
student got a B or better.
You!
1)
Construct the probability distribution for
the number of boys in a three-child family.
Find the following probabilities:
P(2 or more boys)
2)
P(No boys)
3)
P(1 or less boys)
In an article in the journal Developmental Psychology (March 1986), a probability
distribution for the age X (in years) when male college students began to
shave regularly is shown:
Here is the probability distribution for X in table form:
X
11
12 13
14
15
16
17
18
19
20+
P(x)
0.013
0
0.067
0.213
0.267
0.240
0.093
0.067
0.013
1)
2)
3)
0.027
Is this a valid probability distribution? What is the random variable of interest?
Is X discrete?
What is the most common age at which a randomly selected male college
student begins shaving?
What is the probability that a randomly selected male college student begins
shaving at 16? What is the probability that a randomly selected male college
student begins shaving before 15?
Continuous Random Variables
.
Example:
S = {all numbers x between 0 and
1 inclusive}
The probability distribution of X
assigns probabilities as area
under a density curve
Any density curve has area
exactly 1 underneath it
(probability = 1)
Example
A random number generator
will spread its output uniformly
across the entire interval from
0 to 9 as we allow it to
generate a long sequence of
numbers. The results of many
trials are represented by the
density curve of a uniform
distribution.
Find the probability that
the generator produces a
number X between 3 and
7
Find the probability that
the generator produces a
number X less than or
equal to 5 or greater
than 8
Special Note:
All continuous probability distributions assign
probability 0 to every individual outcome.
The probability of x >.8 is the same as x ≥ .8
Example:
Find P(.79 < x < .81) =
Find P(.799 < x < .801) =
Find P(.7999 < x < .8001) =
Find P(x=.8) =
Normal Distributions as
Probability Distributions
Because any density curve describes an
assignment of probabilities, normal distributions
are probability distributions.
If X has the N( , ) distribution, then
z
x
is a standard normal random variable having the
distribution N(0,1).
Example
An opinion poll asks an
SRS of 1500 adults what
they consider to be the
most serious problem
affecting schools. Suppose
that if we could ask all
adults this ?, 30% would
say “drugs.”
Assume your sample
proportion follows a
normal distribution: N(.3,
.0118).
Given: Mean = .3, and Standard
dev. = .0118
Find the probability that the poll
result differs from the truth about
the population by more than 2
percentage points.
1) The probabilities that a randomly selected customer purchases 1, 2, 3,
4, or 5 items at a convenience store are .32, .12, .23, .18, and .15,
respectively.
a) Identify the random variable of interest. X = ____. Then
construct a probability distribution (table), and draw a probability
distribution histogram.
b) Find P(X>3.5)
c) Find P(1.0 <X<3.0)
d) Find P(X<5)
2) A certain probability density function is made up of two straight-line
segments. The first segment begins at the origin and goes to the
point (1,1). The second segment goes from (1,1) to the point (x, 1).
a) Sketch the distribution function, and determine what x has to be
in order to be a legitimate density curve.
b) Find P(0<X<.5)
c) Find P(X=1)
d) Find P(0<X<1.25)
e) Circle the correct option: X is an example of a (discrete)
(continuous) random variable.