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4.4
Means and Variances of Random Variables
Mean of a discrete random variable
Suppose that X is a discrete random variable whose distribution is
Value of X
x
1
x
2
x
3
…
x
k
Probability
p
1
p
2
p
3
…
p
k
To find the mean of X, multiply each possible value by its probability, then add all the
products.
μ
x
=
x
1
p
1
+ x
2
p
2
+ x
3
p
3
+
…+
x
k
p
k
μ
x
=Σ x
i
p
i
the mean of a random variable is also frequently referred to as the expected value
of a
random variable:
E(X)
E(X) = μ
x
=Σ x
i
p
i
Example
lottery choose a 3digit number (000999)
probability 1/1000 of winning $500
X is a random variable the amount your ticket pays
probability distribution of X:
Payoff X
$0
$500
Probability
0.999
0.001
What is the average outcome from many tickets (longrun average payoff)?
●play lottery several times→ call the mean of the actual amounts you win
x
●longrun average outcome= mean of random variable = μ
x
Figure 4.13
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View Full DocumentStatistical estimation and the law of large numbers
The population distribution
of a variable is the distribution of its values for all members of the
population.
The population distribution is also the probability distribution (sampling distribution
)
of the variable when we choose one individual at random from the population.
sampling distribution
of a statistic is the distribution of values taken by the statistic in all
possible samples of the same size from the same population
EX
consider the heights of young women
suppose the population = 1000/ then suppose take SRS of size n = 1
population distribution→ 1000 possible values
sampling distribution→ 1000 possible samples of size 1 from population of 1000
sampling distribution→ 1000 possible values
The distribution of heights of young women (1824) is N(64.5, 2.5).
Select a young woman at random and measure her height (random variable X).
In repeated sampling X will have the same N(64.5, 2.5) distribution.
population distribution= sampling distribution when we choose one individual at random
from population
We want to estimate the mean height (μ) of all American women (1824).
This μ = μ
x
of the random variable X obtained by choosing a young woman at random and
measuring her height.
To estimate μ, we choose an SRS of young women and use the sample mean
x
to estimate the
unknown population mean μ.
μ = parameter
x
= statistic
We don’t expect
x
to be exactly equal to μ, and we realize that if we choose another SRS, the
luck of the draw will probably produce a different
x
.
If
x
is rarely exactly right and varies from
sample to sample, why is it nonetheless a reasonable estimate of the population mean μ?
1)
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 Spring '11
 Bill
 Probability, Variance

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