random variables
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Random Variables. Probability Continued Chapter 7. Random Variables. - PowerPoint PPT PresentationTRANSCRIPT
Random Variables
Probability Continued
Chapter 7
Random Variables
Suppose that each of three randomly selected customers purchasing a hot tub at a certain store chooses either an electric (E) or a gas (G) model. Assume that these customers make their choices independently of one another and that 40% of all customers select an electric model. The number among the three customers who purchase an electric hot tub is a random variable. What is the probability distribution?
Random Variable Example
XP(X)
X = number of people who purchase electric hot tub
0 1 2 3
GGG (.6)(.6)(.6)
.216
EGGGEGGGE
(.4)(.6)(.6)(.6)(.4)(.6)(.6)(.6)(.4)
.432
EEGGEEEGE
(.4)(.4)(.6)(.6)(.4)(.4)(.4)(.6)(.4)
.288
EEE (.4)(.4)(.4)
.064
Random Variables
A numerical variable whose value depends on the outcome of a chance experiment is called a random variable.discrete versus continuous
Discrete vs. Continuous
The number of desks in a classroom.
The fuel efficiency (mpg) of an automobile.
The distance that a person throws a baseball.
The number of questions asked during a statistics final exam.
Discrete versus Continuous Probability Distributions
Which is which?
Properties:For every possible x value, 0 < x < 1.Sum of all possible probabilities add to 1.
Properties:Often represented by a graph or function.Area of domain is 1.
Probability Histograms
We can create a probability histogram to show the distributions of discrete random variables.
Example
Let X represent the sum of two dice.
Then the probability distribution of X is as follows:
X 2 3 4 5 6 7 8 9 10 11 12P(X) 1
36236
3 36
436
536
636
536
436
336
236
1 36
Continuous Random Variable and Density Curves
The probability distribution of a continuous random variable assigns probabilities under a density curve.
Probabilities are assigned to INTERVALS of outcomes rather than to individual outcomes.
A probability of 0 is assigned to every individual outcome in a continuous probability distribution.
The Normal Distribution can be a Probability DistributionThe normal curve
Means and VariancesThe mean value of a random variable X (written x ) describes where the probability distribution of X is centered.
We often find the mean is not a possible value of X, so it can also be referred to as the “expected value.”The standard deviation of a random variable X (written x )describes variability in the probability distribution.
Mean of a Random Variable Example
Below is a distribution for number of visits to a dentist in one year. X = # of visits to the dentist.
Determine the expected value, variance and standard deviation.
0 1 2 3 4
( ) .1 .3 .4 .15 .05
X
P X
Formulas
Mean of a Random Variable
X i ix p
Variance of a Random Variable
2 2( )X i X ix p
Mean of a Random Variable Example
0 1 2 3 4
( ) .1 .3 .4 .15 .05
X
P X
E(X) = 0(.1) + 1(.3) + 2(.4) + 3(.15) + 4(.05)
= 1.75 visits to the dentist
X i ix p
Variance and Standard Deviation of a Random Variable Example
0 1 2 3 4
( ) .1 .3 .4 .15 .05
X
P X
Var(X) = (0 – 1.75)2(.1) + (1 – 1.75)2(.3) + (2 – 1.75)2(.4) + (3 – 1.75)2(.15) + (4 – 1.75)2(.05) = .9875
2 2( )X i X ix p
.9875 .9937 visitsX