Transcript Mean and Variance of a Random Variable

Special Topics
Mean of a Probability Model
 The mean
x
of a set of observations is the ordinary
average.
 The mean x of a probability model is also an average,
but with an essential change, not all outcomes are
equally likely.
 It is actually a weighted average.
Formula for Mean of a Discrete
Probability Model
 If the probability distribution of a probability model
is as follows:
Value of X
x1
x2
x3
xn
Probability
p1
p2
p3
pn
 To find the mean (AKA – expected value), multiply
each possible value by its probability, then add all the
products.
x  x1 p1  x2 p2  ...  xn pn
Calculator Shortcut
EX: The distribution of the count of heads in 4 tosses was found
to be:
x
= 0(.0625) + 1(.25) + 2(.375) + 3(.25) + 4(.0625) = 2
Put X in L1 and P(X) in L2, in L3 (L1 x L2). You then can sum this
total to get the mean or expected value.
You can also run a 1VARS Stats on L1,L2 and it will produce the
expected value.
Standard Deviation of a Discrete Probability
Model
 If the probability distribution of a probability model
is as follows:

Value of X
x1
x2
x3
xn
Probability
p1
p2
p3
pn
 To find the standard deviation of the model:
SD 
 x1  x 
2
p1   x2  x  p2   x3  x  p3 ...   xn  x  pn
2
2
2
Calculator:
 L1: Put the values of the random variable
 L2: Put the probabilities of each value
1
 At this point you can run a VARS Stats : L1, L2
it will give you the expected value and the standard
deviation.
Mean of a Continuous Probability Model
 What about continuous probability models? Think of
the area under a density curve as being cut out of solid
homogenous material. The mean μ is the point at
which the shape would balance. This is what this idea
looks like with a skewed model:
Mean of a Continuous Probability Model
 When the model is symmetric (normal, uniform, or
other symmetric shape), the mean (and the median)
lies at the center of the curve.
Mean
Median
The Law of Large Numbers
The Law of Large Numbers
 The law of large numbers tells us that in many trials the
proportion of trials on which an outcome occurs will always
approach its probability.
 The law of large numbers also explains why gambling can
be a business. The winnings (or losses) of a gambler on a
few plays are uncertain—that's why gambling is exciting. It
is only in the long run that the mean outcome is
predictable. The house plays many tens of thousands of
times. So the house, unlike individual gamblers, can count
on the long-run regularity described by the law of large
numbers.
Homework
 Worksheet 8.5