Transcript Section 7.1 - Cabarrus County Schools

Section 6.1
Discrete and Continuous Random
Variables
Sample Spaces
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In Chapter 6, we studied sample spaces.
If we toss two coins, the sample space is
{HH, HT, TH, or TT}.
We statisticians like numbers. So, let’s
convert this sample space to numbers by
counting the number of heads. Then the
sample space is {0, 1, 2}.
Random Variables
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If we let X = the number of heads, then X
is a random variable.
Each time we perform a trial, we’d get
different values for X.
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A random variable is a variable whose value is
a numerical outcome of a random
phenomenon.
Discrete and Continuous Random
Variables
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There are two types of random variables:
discrete and continuous.
Discrete random variables have a finite
(countable) number of possible values.
We use a histogram to graph a discrete random variable.
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Continuous random variables take on all
values in an interval of numbers.
We use a density curve to graph continuous random
variables.
Discrete Random Variables
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Since discrete random variables have a countable
number of outcomes, we can list each outcome
and its probability in a table. This is called a
probability distribution. x represents the random
variable, and p(x) is the probability a random
variable occurs.
x
0
1
2
p(x)
¼
½
¼
Notice the probabilities are all between 0 and 1.
The sum of the probabilities = 1.
The rules of probability are still the same!
Histogram for the coins
Probability
Probability Distribution of Tossing
Two Coins
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
Number of Heads
Grades
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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 the class.
The student’s grade on a 4-point scale (A=4.0)
is a random variable x from 0 - 4.
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Write the probability distribution of x.
Find the probability that the student has an A or a B.
Find P(2 < x < 4).
Find P(2 ≤ x ≤ 4).
Graph the probability distribution and describe the
graph.
Toss 4 Coins
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Let x = the number of Heads in 4 coin
tosses.
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List all the possible outcomes of four different
tosses.
Write the probability distribution of x.
Graph the probability distribution. Describe
the graph.
Find the probability of tossing at least two
heads.
Find the probability of tossing at least one
head.
Continuous Random Variables
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Continuous Random Variables are over an
interval, in which we cannot count all of the
possible outcomes.
To find probabilities of continuous random
variables, we use the area under the density
curve.
Because there are infinitely many possibilities,
we cannot assign probabilities to each value of
x.
Recall that the area under a density curve has a
total area of 1. It is always positive. Sound
familiar???
Common Types of Continuous
Random Variables
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Uniform Distribution – Has the same height
throughout
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Choose a number at random between 0 and 1.
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Find P(0.3 ≤ X ≤ 0.7)
Find P(X ≤ 0.5)
Find P(X > 0.8)
Find P(X ≤ 0.5 or X > 0.8)
Note: Be sure that the area under the curve is 1.
This will help you find the height.
Note: < or ≤ are essentially the same with
continuous random variables, as an individual number
has no area.
Common Types of Continuous
Random Variables
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Normal Distribution
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Suppose X is distributed normally with a mean
of 32 and a standard deviation of 2.
Find P(X > 35).
 Find P(31 ≤ X ≤ 35).
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More on the Normal Distribution
An opinion poll asks an SRS of 1500 American
adults what they consider to be the most serious
problem facing our schools. Suppose that if we
could ask ALL adults this question, 30% would
say “drugs.” This random variable is denoted:
p̂ ~ N(0.3, 0.0118), where p̂ is the predicted
probability.
Find the probability that the poll results differ from
the truth about the population by more than two
percentage points?
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Key Point to Remember
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All continuous probability distributions
assign probability 0 to every individual
outcome.
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So, P(x = 0.8) in a uniform distribution is 0.
P(0.5 < x < 0.8) = P(0.5 ≤ X ≤ 0.8)
We can ignore the distinction between < and
≤ in continuous random variables.
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THE SAME IS NOT TRUE FOR DISCRETE RANDOM
VARIABLES!!!
Homework
Chapter 6
#2, 4, 6, 8, 21, 24