Discrete Probability

Question 1

random variable

Answer 1

is a variable whose values are determined by chance.

Question 2

discrete probability distribution

Answer 2

consists of the values a random variable can
assume and the corresponding probabilities of the values. The probabilities are
determined theoretically or by observation.

Question 3

formula for the mean of the probability distribution

Answer 3

The mean of a random variable with a discrete probability distribution is
m X1 P(X1) X2 P(X2) X3 P(X3) Xn P(Xn)
X P(X)

where X1, X2, X3, . . . , Xn are the outcomes and P(X1), P(X2), P(X3), . . . , P(Xn) are the corre-
sponding probabilities.

Note: X P(X) means to sum the products.

Question 4

formula for variance

Answer 4

Find the variance of a probability distribution by multiplying the square of each outcome by
its corresponding probability, summing those products, and subtracting the square of the
mean. The formula for the variance of a probability distribution is
s2 [X2 P(X)] m2
The standard deviation of a probability distribution is

or s 2[X2 P1X2] m2 s 2s2
Remember that the variance and standard deviation cannot be negative.

Question 5

formula for standard deviation

Answer 5

For a probability distribution, the mean of the random variable describes the measure of the
so-called long-run or theoretical average, but it does not tell anything about the spread of

the distribution. Recall from Chapter 3 that to measure this spread or variability, statisti-
cians use the variance and standard deviation. These formulas were used:

or

These formulas cannot be used for a random variable of a probability distribution since N
is infinite, so the variance and standard deviation must be computed differently.

Question 6

expected value

Answer 6

of a discrete random variable of a probability distribution is the
theoretical average of the variable. The formula is
m E(X) X P(X)
The symbol E(X) is used for the expected value.

Question 7

binomial experiment

Answer 7

experiment is a probability experiment that satisfies the following four
requirements:
1. There must be a fixed number of trials.
2. Each trial can have only two outcomes or outcomes that can be reduced to two
outcomes. These outcomes can be considered as either success or failure.
3. The outcomes of each trial must be independent of one another.
4. The probability of a success must remain the same for each trial.

Question 8

binomial distribution

Answer 8

The outcomes of a binomial experiment and the corresponding probabilities of these
outcomes

Question 9

notation for binomial distribution