Week 1 - 4.3 Mean And Standard Deviation Of Discrete Random Variable Flashcards
The most important characteristics of any probability distribution are the mean (or average value) and the standard
deviation (a measure of how spread out the values are). The example below illustrates how to calculate the mean
and the standard deviation of a random variable.
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A common symbol for the mean is
µ (mu), the lowercase m of the Greek alphabet.
A common symbol for standard deviation is
s (sigma), the Greek lowercase s.
Example: Recall the probability distribution of the 2-coin experiment. Calculate the mean of this distribution.
If we look at the graph of the 2-coin toss experiment (shown below), we can easily reason that the mean value is
located right in the middle of the graph, namely, at x = 1. This is intuitively true. Here is how we can calculate it:
To calculate the population mean, multiply each possible outcome of the random variable X by its associated
probability and then sum over all possible values of X:
Figure on Module
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The mean value, or expected value, of a discrete random variable X is given by the following equation:
µ = E(x) = Âxp(x)
Example: An insurance company sells life insurance of $15,000 for a premium of $310 per year. Actuarial tables
show that the probability of death in the year following the purchase of this policy is 0.1%. What is the expected
gain for this type of policy?
There are two simple events here: either the customer will live this year or will die. The probability of death, as
given by the problem, is 0.001, and the probability that the customer will live is 10.001 = 0.999. The company’s
expected gain from this policy in the year after the purchase is the random variable, which can have the values shown
in the table below.
Table 4.4 on module
Figure: Analysis of the possible outcomes of an insurance policy.
Remember, if the customer lives, the company gains $310 as a profit. If the customer dies, the company “gains”
$310$15,000 = $14,690, or in other words, it loses $14,690. Therefore, the expected profit can be calculated
as follows:
Solution on module
This tells us that if the company were to sell a very large number of the 1-year $15,000 policies to many people, it
would make, on average, a profit of $295 per sale
Another approach is to calculate the expected payout, not the expected gain:
solution on module
Since the company charges $310 and expects to pay out $15, the average profit for the company is $295 per policy
Sometimes, we are interested in measuring not just the expected value of a random variable, but also the variability
and the central tendency of a probability distribution. To do this, we first need to define population variance, or s2.
It is the average of the squared distance of the values of the random variable X from the mean value, µ. The formal
definitions of variance and standard deviation are shown below
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The variance of a discrete random variable is given by the following formula:
s2 = Â(xµ)2P(x)
The square root of the variance, or, in other words, the square root of s2, is the standard deviation of a discrete
random variable:
Formula on Module
Example: A university medical research center finds out that treatment of skin cancer by the use of chemotherapy
has a success rate of 70%. Suppose five patients are treated with chemotherapy. The probability distribution of x
successful cures of the five patients is given in the table below:
Table on Module
Figure: Probability distribution of cancer cures of five patients.
a) Find µ.
b) Find s.
c) Graph p(x) and explain how µ and s can be used to describe p(x).
a. To find µ, we use the following formula:
formula on module
b. To find s, we first calculate the variance of X:
Solution on module
Now we calculate the standard deviation:
solution on module
We can use the mean, or µ, and the standard deviation, or s, to describe p(x) in the same way we used x and s to
describe the relative frequency distribution. Notice that µ = 3.5 is the center of the probability distribution. In other
words, if the five cancer patients receive chemotherapy treatment, we expect the number of them who are cured to be
near 3.5. The standard deviation, which is s = 1.02 in this case, measures the spread of the probability distribution
p(x).
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The mean value, or expected value, of the discrete random variable X is given by
µ = E(x) = Â xp(x)
The variance of the discrete random variable X is given by
S2 = Â(xµ)2 p(x
The square root of the variance, or, in other words, the square root of s2, is the standard deviation of a discrete
random variable:
s = ps2.
