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Chapter 7. Discrete Distributions. Random Variable -. A numerical variable whose value depends on the outcome of a chance experiment. Two types:. Discrete – count of some random variable Continuous – measure of some random variable. Discrete Probability Distribution. - PowerPoint PPT Presentation
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Chapter 7Chapter 7Discrete
Distributions
Random Variable Random Variable --•A numerical variable whose value depends on the outcome of a chance experiment
Two types:Two types:•Discrete – count of some random variable
•Continuous – measure of some random variable
Discrete Probability Discrete Probability DistributionDistribution
•Gives the values associated with each possible x value
•Usually displayed in a table, but can be displayed with a histogram or formula
Properties for a Properties for a discrete probability discrete probability
distributiondistribution
1)For every possible x value, 0 < P(x) < 1.
2) For all values of x,
P(x) = 1.
Suppose you toss 3 coins & record Suppose you toss 3 coins & record the number of heads.the number of heads.
The random variable X defined as ...
Create a probability distribution.
Create a probability histogram.
The number of heads tossed
X 0 1 2 3
P(X) .125 .375 .375 .125
Let x be the number of courses for which a randomly selected student at a certain university is registered.
X 1 2 3 4 5 6 7 P(X) .02 .03 .09 ? .40 .16 .05
P(x = 4) =
P(x < 4) =
P(x < 4) =
What is the probability that the student is registered for at least five courses?
Why does this not start at zero?
.25
.14
.39 P(x > 5) = .61
Formulas for mean & Formulas for mean & variancevariance
ixix
iix
px
px
22
Found on formula card!
Let x be the number of courses for which a randomly selected student at a certain university is registered.
X 1 2 3 4 5 6 7
P(X) .02 .03 .09 .25 .40 .16 .05
What is the mean and standard deviations of this distribution?
4.66 & = 1.2018
Here’s a game:
If a player rolls two dice and gets a sum of 2 or 12, he wins $20. If he gets a 7, he wins $5. The cost to roll the dice one time is $3. Is this game fair?
A fair game is one where the cost to play EQUALS the expected
value!X 0 5 20
P(X) 7/9 1/61/18
NO, since = $1.944 which is less than it cost to play ($3).
Linear function of a Linear function of a random variablerandom variable
If x is a random variable and a and b are numerical constants, then the random variable y is defined by
• andbxay
xyxbxay
xbxay
bb
ba
or2222
The mean is changed by addition &
multiplication! The standard deviation is ONLYONLY
changed by multiplication!
Let x be the number of gallons required to fill a propane tank. Suppose that the mean and standard deviation is 318 gal. and 42 gal., respectively. The company is considering the pricing model of a service charge of $50 plus $1.80 per gallon. Let y be the random variable of the amount billed. What is the mean and standard deviation for the amount billed? = $622.40 & = $75.60
Linear combinationsLinear combinations
222
21
21y
21
...
...
then...If
xnxxy
xnxx
nxxxy
Just add or subtract the means!
If independent, always addadd the variances!
A nationwide standardized exam consists of a multiple choice section and a free response section. For each section, the mean and standard deviation are reported to be
mean SD
MC 38 6
FR 30 7
If the test score is computed by adding the multiple choice and free response, then what is the mean and standard deviation of the test?
68 & = 9.2195