©Zarestky
Math 141 Week in Review
Texas A&M University
Week 8 Key Topics
8.1 Distributions of Random Variables
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•
•
•
A random variable is a rule that describes the outcome of an experiment.
A finite discrete random variable assumes only finitely many values.
An infinite discrete random variable assumes infinitely many values that may be arranged in
a sequence.
A continuous random variable assumes values that comprise an interval of the real numbers.
8.2 Expected Value
Histograms and Measures of Central Tendency/Dispersion
• The mean, denoted x , is where the histogram “balances.”
• The mode is the tallest rectangle. This is the most frequently occurring value.
• The median is where the area is cut in half. This is the middle value.
• The range is the number of rectangles. (Remember, some may have a height of 0.)
Expected Value
• The expected value for the finite discrete variable X in a probability distribution is:
E(X) = x1p1 + x2p2 + ... + xnpn
• It is equal to the mean.
• Expected value is used for life insurance premiums, lotteries, raffles, expected winnings,
expected profits, etc.
• Organize the values and probabilities in a table to find the expected value more easily.
Odds
•
If P(E) is the probability of event E occurring, then the odds in favor of E are
P (E)
P (E)
where P ( E ) ! 1 .
=
1 ! P (E ) P EC
( )
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•
•
If P(E) is the probability of an event occurring, then the odds against E occurring are
C
1! P (E) P E
=
where P ( E ) ! 0 .
P (E)
P (E)
a
If the odds in favor of an event E occurring are a to b, then P ( E ) =
.
a+b
Express the odds as a ratio of whole numbers: a to b, or a:b
( )
1
©Zarestky
Math 141 Week in Review
Texas A&M University
8.3 Variance and Standard Deviation
Variance
• Variance is a measure of distance from the mean.
• Suppose a random variable has expected value E(X) = µ and probability distribution
x
x1
x2
x3
...
xn
P(X = x)
p1
p2
p3
...
pn
Then the variance of the random variable X is:
Var ( X ) = p1 ( x1 ! µ ) + p2 ( x2 ! µ ) + ... + pn ( xn ! µ )
2
•
2
2
The units are the square of the units of the random variable.
Standard Deviation
• The standard deviation, σ, describes how far away from the mean one can expect the
outcomes to be.
•
! = Var ( X )
•
The units are the same as the random variable.
2