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Chapter 7
• We all know that random experiment generates random variable that
have inherent uncertainty.
• All random variables are subject to chance
• This chapter concerns about the number of events of particular type.
For example, we might believe that 99% of all circuit relays are
acceptable. If so, how many acceptables will be identified in 100
relays inspected?
• In this case we shall see, there is no single answer to these
questions
• Some quantities are more likely to occur the others, but many
results are possible.
• We shall see how to compute probabilities for each outcome.
• Collectively, the possible values for a random variable and their
associated probability value establish a probabilities distribution.
• Probability distributions may involve random variables that assume
discrete values.
• They may instead involve values that can fall anywhere within a
specified range of real numbers
EXPECTED VALUE AND VARIANCE
• It’s important to compute mean (expected value) and
variance of probability distribution. For example,
– Recall from our discussion on random variables and
random numbers that if we want to generate random
numbers, it may be necessary to specify mean and
variance (along with the distribution) of the random
numbers.
– Suppose that you have to decide whether or not to
make an investment that has an uncertain return. You
may like to know whether the expected return is more
than the investment.
EXPECTED VALUE
• The expected value of a random variable is simply a
weighted average of the possible values, using the
respective probabilities as weights.
• The expected value of a discrete random variable is
obtained as follows:
n
E  X    xi pxi 
i 1
• E(X) is the expected value of the random variable X
• xi is the i-th possible value of the random variable X
• p(xi) is the probability that the random variable X will
assume the value xi
Expected Value
• The expected value measures the random
variable’s central tendency,
• E(X) is often referred to as the mean of X.
• The usual interpretation of an expected
value is that it represents the long-run
average result from a series of repeated
random experiments.
Example from text book
Possible Numbers of
Failures y
Probability p(y)
Weighted Value yp(y)
0
.1353
0
1
.2707
.2707
2
.2707
.5414
3
.1804
.5412
4
.0902
.3608
5
.0361
.1805
6
.0121
.0726
7
.0034
.0238
8
.0009
.0072
9
.0002
.0018
E(Y) = 2.000
Example from the text book
• The expected value here is 2.0, indicating
that the mean number of failures is 2 per
month.
• Over several years of similar operation the
actual tally of month-by-month failures
should average to nearly 2
Variance
• The common measure of dispersion for a
random variable is its variance
• This measure is also a weighted average,
wherein the quantities involved indicate
how much individual values differ from the
center of the distribution.
• These quantities are the squared
deviations of each possible level from the
expected value.
Variance
• The variance is an important statistical concept
because it provides a systematic summary of
individual differences and because of its convenient
mathematical properties.
• In general, we will use the following expression.
• Variance for discrete random variable
Var( X )  [ x  E( X )] p( x)
2
Variance
• Here, as before, the summation is taken
over all possible levels for the random
variable.
• Continuing with the example of number of
equipment failures Y, next table shows the
variance computation.
Example
Possible
Numbers of
Failures y
Probability p(y)
0
.1353
1
.2707
2
.2707
3
.1804
4
.0902
5
.0361
6
.0121
7
.0034
8
.0009
9
.0002
Squared deviation
[y-E(Y)]2
Some Important Properties of Expected value
• Var(X) is the expected value of a function of
X:
[ X  E( X )]2
In general, E[g(X)]   g(x)p(x)
We can Apply the above for variance. Note that the variance is itself the expected value of a function of X.
2
Var( X )  [ X  E( X )] p( x)  E([ X  E( X )]2)
Some Important Properties of Expected
value
• Three of the key properties of expected value are listed.:
1. E c   c
2. E cX   cE  X 
3. E (a  bX )  a  bE ( X )
• X is random variables
• c is a constant
• E(X), and E(c) are expected values of X and c
respectively.
Two Important properties of the variance are
listed
• The laws of expected value are listed below:
1. V c   0
2. Var a  bX   b 2Var  X 
• X and Y are random variables
• c is a constant
• V(X), V(Y), and V(c) are variances of X, Y and c
respectively.
Expected value and Variance of a
continuous Random variable
• Recall that the expected value of a discrete
random variable expresses its central tendency.
It is a weighted average by summing the
products of each possible level times the
respective probability
• The expected value of a continuous random
variable is found analogous by integrating the
product of the dummy variable and the density
over the entire span of possibilities.
Expected value of a continuous
random variable

E( X )   xf ( x)dx

Expected value of a continuous
random variable
• As an example, consider the distance X
between magnetic tape flaws in the
preceding example. Because f(x)=0 for all
x>0 the lower of integration will be taken
as 0.