5 random variables
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Transcript of 5 random variables
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1Slide
Random VariablesDefinition & Example
Definition: A random variable is a quantity resulting from a random experiment that, by chance, can assume different values.
Example: Consider a random experiment in which a coin is tossed three times. Let X be the number of heads. Let Hrepresent the outcome of a head and T the outcome of a tail.
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2Slide
The sample space for such an experiment will be: TTT, TTH, THT, THH, HTT, HTH, HHT, HHH.
Thus the possible values of X (number of heads) are X = 0, 1, 2, 3.
This association is shown in the next slide.
Note: In this experiment, there are 8 possible outcomes in the sample space. Since they are all equally likely to occur, each outcome has a probability of 1/8 of occurring.
Example (Continued)
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3Slide
TTT
TTH
THT
THH
HTT
HTH
HHT
HHH
0
1
1
2
1
2
2
3
Sample
Space
X
Example (Continued)
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4Slide
The outcome of zero heads occurred only once.
The outcome of one head occurred three times.
The outcome of two heads occurred three times.
The outcome of three heads occurred only once.
From the definition of a random variable, X as defined in this experiment, is a random variable.
X values are determined by the outcomes of the experiment.
Example (Continued)
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5Slide
Let x = number of TVs sold at the store in one day,
where x can take on 5 values (0, 1, 2, 3, 4)
Example: JSL Appliances
Discrete random variable with a finite number of values
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6Slide
Let x = number of customers arriving in one day,
where x can take on the values 0, 1, 2, . . .
Example: JSL Appliances
Discrete random variable with an infinite sequence of values
We can count the customers arriving, but there is nofinite upper limit on the number that might arrive.
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7Slide
Probability Distribution: Definition
Definition: A probability distribution is a listing of all the outcomes of an experiment and their associated probabilities.
The probability distribution for the random variable X (number of heads) in tossing a coin three times is shown next.
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8Slide
Probability Distribution for Three Tosses of a Coin
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9Slide
RANDOM VARIABLE
.10
.20
.30
.40
0 1 2 3 4
Random Variables
Discrete Probability Distributions
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10Slide
Discrete Random Variable Examples
Experiment Random
Variable
Possible
Values
Make 100 sales calls # Sales 0, 1, 2, ..., 100
Inspect 70 radios # Defective 0, 1, 2, ..., 70
Answer 33 questions # Correct 0, 1, 2, ..., 33
Count cars at toll
between 11:00 & 1:00
# Cars
arriving0, 1, 2, ...,
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11Slide
The probability distribution for a random variabledescribes how probabilities are distributed overthe values of the random variable.
We can describe a discrete probability distributionwith a table, graph, or equation.
Discrete Probability Distributions
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12Slide
The probability distribution is defined by aprobability function, denoted by f(x), which providesthe probability for each value of the random variable.
The required conditions for a discrete probabilityfunction are:
Discrete Probability Distributions
f(x) > 0
f(x) = 1
P(X) ≥ 0ΣP(X) = 1
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13Slide
a tabular representation of the probabilitydistribution for TV sales was developed.
Using past data on TV sales, …
Number
Units Sold of Days
0 80
1 50
2 40
3 10
4 20
200
x f(x)
0 .40
1 .25
2 .20
3 .05
4 .10
1.00
80/200
Discrete Probability DistributionsExample
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14Slide
.10
.20
.30
.40
.50
0 1 2 3 4Values of Random Variable x (TV sales)
Pro
bab
ilit
y
Discrete Probability Distributions
Graphical Representation of Probability Distribution
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15Slide
Discrete Probability Distributions
As we said, the probability distribution of a discrete random variable is a table, graph, or formula that gives the probability associated with each possible value that the variable can assume.
Example : Number of Radios Sold at Sound City in a Weekx, Radios p(x), Probability0 p(0) = 0.031 p(1) = 0.202 p(2) = 0.503 p(3) = 0.204 p(4) = 0.055 p(5) = 0.02
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16Slide
Expected Value of a Discrete Random Variable
The mean or expected value of a discrete random
variable is:
xAll
X xxp )(
Example: Expected Number of Radios Sold in a Week
x, Radios p(x), Probability x p(x)
0 p(0) = 0.03 0(0.03) = 0.00
1 p(1) = 0.20 1(0.20) = 0.20
2 p(2) = 0.50 2(0.50) = 1.00
3 p(3) = 0.20 3(0.20) = 0.60
4 p(4) = 0.05 4(0.05) = 0.20
5 p(5) = 0.02 5(0.02) = 0.10
1.00 2.10
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17Slide
Variance and Standard Deviation
The variance of a discrete random variable is:
xAll
XX xpx )()( 22
2
XX
The standard deviation is the square root of the variance.
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18Slide
Example: Variance and Standard Deviation of the Number of
Radios Sold in a Week
x, Radios p(x), Probability (x - X)2 p(x)
0 p(0) = 0.03 (0 – 2.1)2 (0.03) = 0.1323
1 p(1) = 0.20 (1 – 2.1)2 (0.20) = 0.2420
2 p(2) = 0.50 (2 – 2.1)2 (0.50) = 0.0050
3 p(3) = 0.20 (3 – 2.1)2 (0.20) = 0.1620
4 p(4) = 0.05 (4 – 2.1)2 (0.05) = 0.1805
5 p(5) = 0.02 (5 – 2.1)2 (0.02) = 0.1682
1.00 0.8900
89.02 X
Variance
9434.089.0 X
Standard deviation
Variance and Standard Deviation
µx = 2.10
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19Slide
Expected Value and Variance (Summary)
The expected value, or mean, of a random variableis a measure of its central location.
The variance summarizes the variability in thevalues of a random variable.
The standard deviation, , is defined as the positivesquare root of the variance.
Var(x) = 2 = (x - )2f(x)
E(x) = = xf(x)