Chapter 9: Random Variables and Probability Distributions
9.1 Expected Value of a Random Variable
Random variable: assumes any of several different values as a result of some
o Denoted by a capital letter, such as X.
o The value it may take on can be denoted by the corresponding
lowercase letter, for example, x.
Discrete random variable: a random variable that can take one of a finite
number of distinct outcomes.
o I.e. either 10, 20, 30, 40, or 50 etc.
Continuous random variable: a random variable that can take any numeric
value within a range of values. The range may be infinite or bounded at either
or both ends.
o I.e. 10 < x < 20
Probability distribution: a function that associates a probability P with each
value of a discrete random variable X, denoted P(X = x), or with any interval
values of a continuous random variable.
o Also called a probability model.
Expected value (mean): the expected value of a random variable is its
theoretical long-run average value, the centre of its model. Denoted μ or E(X),
it’s found (if the random variable is discrete) by summing the products of
variable values and probabilities:
( ) ∑ ( )
o It isn’t what actually happens to any particular policyholder; it is the
9.2 Standard Deviation and Variance of a Random Variable
Standard deviation of a (discrete) random variable: describes the spread in
the model and is the square root of the variance.
( ) √ ( ) √ ∑( ( )) ( )
9.3 Adding and Subtracting Random Variables
Changing a random variable by a constant
( ) ( )
( ) ( )
( ) ( ) Adding/subtracting a constant will shift the data but not change the variance
and therefore won’t change the SD.
( ) ( )
( ) ( )
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Addition Rule for Expected Values of Random Variables: the expected value
of the sum (or difference) of random variables is the sum (or difference) of
their expected values.
( ) ( ) ( )
Fundamental principle behind insurance is that by spreading risk among
many policies, a company can keep the SD quite small and predict costs more
Addition Rule for Variances of (Independent) Random Variables: the
variance of the sum or difference of two independent random variables is the
sum of their individual variances.
( ) ( ) ( )
( ) √ ( ) ( )
The variance of one policyholder with an expected payout that is doubled, is
4x the original variance (2 V[X]) whereas the variance of two policyholders
with the original expected payout is just 2x the original variance (V[X] +
o Thus the SD is higher with the one policyholder who has a doubled
possible payout vs. the two policyholders.
o When we subtract the variables, E – I, we add the variances, Var(E) +
o SD’s don’t add or subtract. The SD is obtained by taking the square
root of the variance.
o The expected value of the sum of two random variables is the sum of
the expected values.
o The expected value of the difference of two random variables is the
difference of the expected values.
o If the random variables are independent, the variance of their sum or
difference is always the sum of the variances.
The sum or difference of the variances of correlated variables is:
( ) ( ) ( ) ( ) ( )
The sum or difference of the variances of variables multiplied by constants is:
( ) ( ) ( ) ( ) ( ) The sum or difference of the expected values of the variables multiplied by
( ) ( ) ( )
When r < 0, the variance (riskiness of an investment) is reduced.
o “Don’t put all your eggs in one basket.”
9.4 Discrete Probability Distributions
If X is a random variable with possible outcomes 1, 2, … , n and P(X = i) = 1/n
for each value of I, then we say X has a discrete Uniform Distribution.
By convention, one of the outcomes in a trial is denoted a “success” and the
other a “failure”.
o Often the less common outcome or the one that calls for action is
called a success.
Bernoulli trials: a sequence of trials is called Bernoulli if:
o There are exactly two possible outcomes (usually denoted success
I.e. webpage loads correctly (failure) or does not (success).
o The probability of success is constant.
It is denoted p and the probability of failure is denoted q = 1 –
o The trials are independent.
I.e. the fact that one page loaded correctly will not impact if the
next will or not.
10% condition: we are ok to proceed in using a probability distribution to
model different aspects of Bernoulli trials so long as the sample taken is
smaller than 10% of the population.
o Also requires the outcomes to be independent.
9.6 The Binomial Distribu