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Warm Up: 2003 AP FRQ #2
7.1 Discrete and Continuous
Random Variables
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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
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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
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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!
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1)
2)
3)
Construct the probability distribution
for the number of boys in a three-child
family. Find the following probabilities:
P(2 or more boys)
P(No boys)
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
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 Any density curve has
area exactly 1
underneath it
(probability = 1)
.
Example
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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.
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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:
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All continuous probability distributions
assign probability 0 to every individual
outcome.
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
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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
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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 q, 30% would say
“drugs.”
Assume your sample
proportion follows a
normal distribution:
N(.3, .0118).
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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 straightline 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.