Section 7.1.1 Discrete and Continuous Random Variables
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Transcript of Section 7.1.1 Discrete and Continuous Random Variables
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Section 7.1.1Discrete and Continuous Random VariablesAP Statistics
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AP Statistics, Section 7.1, Part 1 2
Random Variables A random variable is a variable whose value is
a numerical outcome of a random phenomenon. For example: Flip three coins and let X
represent the number of heads. X is a random variable.
We usually use capital letters to denotes random variables.
The sample space S lists the possible values of the random variable X.
We can use a table to show the probability distribution of a discrete random variable.
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AP Statistics, Section 7.1, Part 1 3
Discrete Probability Distribution Table
Value of X: x1 x2 x3 … xn
Probability:p1 p2 p3 … pn
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AP Statistics, Section 7.1, Part 1 4
Discrete Random Variables
A discrete random variable X has a countable number of possible values. The probability distribution of X lists the values and their probabilities.
X: x1 x2 x3 … xk
P(X): p1 p2 p3 … pk
1. 0 ≤ pi ≤ 1
2. p1 + p2 + p3 +… + pk = 1.
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AP Statistics, Section 7.1, Part 1 5
Probability Distribution Table: Number of Heads Flipping 4 Coins
TTTT
TTTH
TTHT
THTT
HTTT
TTHH
THTH
HTTH
HTHT
THHTHHTT
THHH
HTHH
HHTH
HHHT
HHHH
X 0 1 2 3 4
P(X) 1/16 4/16 6/16 4/16 1/16
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AP Statistics, Section 7.1, Part 1 6
Probabilities: X: 0 1 2 3 4 P(X): 1/16 1/4 3/8 1/4 1/16
.0625 .25 .375 .25 .0625 Histogram
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AP Statistics, Section 7.1, Part 1 7
Questions.
Using the previous probability distribution for the discrete random variable X that counts for the number of heads in four tosses of a coin. What are the probabilities for the following?
P(X = 2) P(X ≥ 2) P(X ≥ 1)
.375
.375 + .25 + .0625 = .6875
1-.0625 = .9375
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AP Statistics, Section 7.1, Part 1 8
What is the average number of heads?
61 4 4 116 16 16 16 16
0 4 12 12 416 16 16 16 16
3216
0 1 2 3 4
2
x
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AP Statistics, Section 7.1, Part 1 9
Continuous Random Varibles Suppose we were to randomly generate a
decimal number between 0 and 1. There are infinitely many possible outcomes so we clearly do not have a discrete random variable.
How could we make a probability distribution? We will use a density curve, and the probability
that an event occurs will be in terms of area.
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AP Statistics, Section 7.1, Part 1 10
Definition:
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.
All continuous random distributions assign probability 0 to every individual outcome.
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AP Statistics, Section 7.1, Part 1 11
Distribution of Continuous Random Variable
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AP Statistics, Section 7.1, Part 1 12
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AP Statistics, Section 7.1, Part 1 13
Example of a non-uniform probability distribution of a continuous random variable.
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AP Statistics, Section 7.1, Part 1 14
Problem
Let X be the amount of time (in minutes) that a particular San Francisco commuter must wait for a BART train. Suppose that the density curve is a uniform distribution.
Draw the density curve. What is the probability that the wait is
between 12 and 20 minutes?
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AP Statistics, Section 7.1, Part 1 15
Density Curve.
20151050
0.05
0.04
0.03
0.02
0.01
0.00
X
Densi
ty
Distribution PlotUniform, Lower=0, Upper=20
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AP Statistics, Section 7.1, Part 1 16
Probability shaded.
0.05
0.04
0.03
0.02
0.01
0.00
X
Densi
ty
12
0.4
200
Distribution PlotUniform, Lower=0, Upper=20
P(12≤ X ≤ 20) = 0.5 · 8 = .40
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AP Statistics, Section 7.1, Part 1 17
Normal Curves
We’ve studied a density curve for a continuous random variable before with the normal distribution.
Recall: N(μ, σ) is the normal curve with mean μ and standard deviation σ.
If X is a random variable with distribution N(μ, σ), then is N(0, 1)X
Z
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AP Statistics, Section 7.1, Part 1 18
Example Students are reluctant to report cheating by
other students. A sample survey puts this question to an SRS of 400 undergraduates: “You witness two students cheating on a quiz. Do you go to the professor and report the cheating?”
Suppose that if we could ask all undergraduates, 12% would answer “Yes.” The proportion p = 0.12 would be a parameter for the population of all undergraduates.
p̂
0.016). N(0.12, ofon distributi a with variablerandom a is ˆ . estimate toused
statistic a is yes""answer whosample theof ˆ proportion The
pp
p
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AP Statistics, Section 7.1, Part 1 19
Example continued
Students are reluctant to report cheating by other students. A sample survey puts this question to an SRS of 400 undergraduates: “You witness two students cheating on a quiz. Do you go to the professor and report the cheating?”
What is the probability that the survey results differs from the truth about the population by more than 2 percentage points?
Because p = 0.12, the survey misses by more than 2 percentage points if .14.0ˆor 10.0ˆ pp
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AP Statistics, Section 7.1, Part 1 20
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AP Statistics, Section 7.1, Part 1 21
Example continued Calculationsˆ ˆ ˆ( 0.10 or 0.14) 1 (0.10 0.14)
From Table A,
ˆ0.10 0.12 0.12 0.14 0.12ˆ(0.10 0.14)
0.016 0.016 0.016
( 1.25 1.25)
0.8944 0.1056 0.7888
So,
ˆ ˆ( 0.10 or 0.14) 1 0.7888 0.2112
P p p P p
pP p P
P Z
P p p
About 21% of sample results will be off by more than two percentage points.
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AP Statistics, Section 7.1, Part 1 22
Summary
A discrete random variable X has a countable number of possible values.
The probability distribution of X lists the values and their probabilities.
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.
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AP Statistics, Section 7.1, Part 1 23
Summary
When you work problems, first identify the variable of interest.
X = number of _____ for discrete random variables.
X = amount of _____ for continuous random variables.