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Probability(C14-C17 BVD)
C16: Random Variables
* Random Variable – a variable that takes numerical
values describing the outcome of a random process.
* Probability Distribution (a.k.a. probability model) –
A table that lists all the outcomes a random
variable can take (sample space) and the associated
probabilities for each outcome. Probabilities must
add to 1.
* Random variable notation: X (capital) for the
variable, x (often with a subscript) for an individual
outcome
* P(X=x) is the probability the variable takes on the
value x (or that outcome x happens).
* Discrete Random Variable – the probability
distribution is finite – you can list all the possible
outcomes.
* The mean of a discrete random variable is called
“Expected value”, because it represents the longrun average, or what you would expect in the
long-run.
* µ = E(x) = Σxipi
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*Variance = σ2 = Σ(xi-µ)2pi
*ALWAYS do calculations using variances,
then take square root at end to get σ
*Calculator
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* These can take all values in an interval – there an
infinite number of possible outcomes.
* Their distributions are density curves such as the
normal model.
* If a normal model is appropriate to describe the
distribution, then you can use z-scores and z-table
to find areas under the curve to represent
probabilities (see Ch 6).
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* If Y = a +bX is a transformation of a random
variable X, then…
* µy = a + bµx
* The mean or Expected value of Y is just the
mean of the original distribution times b added
to a.
* σy2 = b2σx2
* The variance of Y is the variance of X times b2.
The “a” does NOT affect spread.
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*Temperature in a dial-set temperature-
controlled bathtub for babies (X) has a
mean temperature of 34 degrees Celsius
with a standard deviation of 2 degrees
Celsius.
*Convert the mean and standard deviation
to Fahrenheit degrees (F = 9/5C +32)
*µy = a + bµx = 32 + 9/5(34) = 93.2 degrees
Fahrenheit
*σy2 = b2σx2 = 81/25(4) = 12.96
*=> σy = 3.6 degrees Fahrenheit
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* When adding/subtracting two different random
variables X and Y:
* E(X+Y) = E(X) + E(Y)
* E(X-Y) = E(X) – E(Y)
* Var(X+Y) = Var(X) + Var(Y)
* Var(X-Y) = Var(X) + Var(Y)
* Notice! Variances add even when random variables
are being subtracted.
* Remember! Take the square root at the very end to
find standard deviations.
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* Let’s say X is a random variable for the amount of a bet.
X + X may represent two one dollar bets. 2X may
represent a single two dollar bet. Let’s say the expected
winnings from one 1-dollar bet is $-0.25 with a standard
deviation of $0.10.
* Expected values may come out the same either way, but
variances/standard deviations probably won’t!
* µy = a + bµx vs. E(X+Y) = E(X) + E(Y)
* 2(0.25)
vs. 0.25 + 0.25 => 0.5 vs. 0.5
* σy2 = b2σx2
* 22(0.1)2
* => 0.2
vs. Var(X+Y) = Var(X) + Var(Y)
vs. (0.1)2 + (0.1)2 => 0.04 vs. 0.02
vs. 0.141 for standard deviations
See pages 315-318 for examples
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