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The Practice of Statistics (4th Edition) - Starnes, Yates, Moore
Chapter 6: Random Variables
Key Vocabulary:
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random variable
discrete random variable
probability distribution
mean of a random
variable
variance of a random
variable
probability density curve
continuous random
variable
standard deviation
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binomial setting
binomial random variable
binomial distribution
binomial coefficient
binomial probability
linear transformation
normal approximation
geometric setting
geometric distribution
geometric random
variable
Normal approximation
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geometric probability
factorial
expected value
standard deviation
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X
Y
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uniform distribution
6.1 Discrete and Continuous Random Variables (pp.341-352)
1. What is a random variable? A random variable takes on numerical values that describe the
outcomes of some chance process.
2. Define probability distribution. The probability distribution of a random variable gives
its possible values and their probabilities.
3. What is a discrete random variable? A discrete random variable X takes a fixed set of
possible values with gaps between.
4. What are the two requirements for the probability distributions of discrete random
variables?
1. Every probability Pi is a number between 0 and 1.
2. The sum of the probabilities is 1: P1 + P2 + P3 +…= 1.
5. If X is a discrete random variable, what information does the probability distribution of X
give? The probability distribution of a discrete random variable X lists the values xi, and
their probabilities Pi.
6. In a probability histogram what does the height of each bar represent (assuming the width
of each bar is the same)? The probability
7. In a probability histogram, what is the sum of the height of each bar? One
8. What is the mean  X of a discrete random variable X? The mean (μx) of a discrete
random variable X is also an average of the possible values of X, but with an important
change to take into account the fact that not all outcomes may be equally likely.
9. How do you calculate the mean of a discrete random variable? … see # 10
The Practice of Statistics (4th Edition) - Starnes, Yates, Moore
10. How do you calculate the mean of a discrete random variable?
To find the mean (expected value) of X, multiply each possible value by its
probability, then add all the products.
11. Define expected value. What notation is used for expected value? The mean; E(X)
12. Does the expected value of a random variable have to equal one of the possible
values of the random variable? Explain. No. The mean is a long-run average over
many repetitions, therefore having an expected value not equal one of the possible
values is not an unlikely occurrence.
13. Explain how to calculate the variance and standard deviation of a discrete
random variable. The variance is an average of the squared deviation (xi – μx)2 of
the values of the variable X from its mean μx. The standard deviation of X, σx, is
the square root of the variance.
14. Explain the meaning of the standard deviation of a random variable X. The
standard deviation of a random variable X is a measure of how much the values of
the variable tend to vary, on average, from the mean μx.
15. What is a continuous random variable and how is it displayed?
A continuous random variable X takes all values in an interval of numbers. The
probability distribution of X is described by a density curve. The probability of
any event is the area under the density curve and above the values of X that make
up the event.
16. If X is a continuous random variable, how is the probability distribution of X
described? The probability distribution for a continuous random variable assigns
probabilities to intervals of outcomes rather than to individual outcomes.
17. What is the area under a probability density curve equal to?
It is always equal to 1.
18. What is the difference between a discrete random variable and a continuous
random variable? We call Y a continuous random variable because its values
are not isolated numbers but rather an entire interval of numbers. In many cases,
discrete random variables arise from counting something; continuous random
variables often arise from measuring something.
19. If X is a discrete random variable, do P  X  2 and P  X  2 have the same
value? Explain. (Discuss in class)
20. If X is a continuous random variable, do P  X  2 and P  X  2 have the same
value? Explain. (Discuss in class)
The Practice of Statistics (4th Edition) - Starnes, Yates, Moore
21. How is a Normal distribution related to probability distribution? Normal
distributions can be probability distributions as well as descriptions of data. There
is a close connection between a Normal distribution as an idealized description for
data and a Normal probability model
6.2
Transforming and Combining Random Variables
(pp. 358-375)
1. What is the effect on a random variable of multiplying or dividing by a constant?
Multiplying (or dividing) each value of a random variable by a number b:
Multiplies (divides) measures of center and location (mean, median, quartiles,
percentiles) by b. Multiplies (divides) measures of spread (range, IQR, standard
deviation) by |b|. Does not change the shape of the distribution.
2. How does multiplying by a constant effect the variance? The variance is
multiplied by the constant.
3. What is the effect on a random variable of adding or subtracting by a constant?
Adding the same number a (which could be negative) to each value of a random
variable: Adds a to measures of center and location (mean, median, quartiles,
percentiles). Does not change shape or measures of spread (range, IQR, standard
deviation).
4. Define linear transformation. Transforming a random variable by both adding or
subtracting a constant and multiplying or dividing by a constant.
5. What are the effects of a linear transformation on the mean and standard
deviation? If Y = a + bX is a linear transformation of the random variable X, then
the probability distribution of Y has the same shape as the probability distribution
of X. μy = a + bμx. AND σy = |b|σx (since b could be a negative number).
6. Define the mean of the sum of random variables. For any two random variables X
and Y, if T = X + Y, then the expected value of T is E(T) = μT = μX + μY. In
general, the mean of the sum of several random variables is the sum of their
means.
7. What are independent random variables? If knowing whether any event
involving X alone has occurred tells us nothing about the occurrence of any event
involving Y alone, and vice versa, then X and Y are independent random
variables.
8. Define the variance of the sum of independent random variables. What types of
variables does it apply to? For any two independent random variables X and Y, if
T = X + Y, then the variance of T is
In general, the variance of the
sum of several independent random variables is the sum of their variances.
The Practice of Statistics (4th Edition) - Starnes, Yates, Moore
9. When can you add the variances of two random variables? When adding two
independent random variables, variances add. Standard deviations do not add.
10. State the equation for the mean of the difference of random variables? For any two
random variables X and Y, if D = X − Y, then the expected value of D is E(D) =
μD = μX − μY. In general, the mean of the difference of several random variables
is the difference of their means.
11. State the formula for the variance of the difference of random variables.
For any two independent random variables X and Y, if D = X − Y, then the
variance of D is
12. What happens if two independent Normal random variables are combined? Any
sum or difference of independent Normal random variables is also Normally
distributed. The mean and standard deviation of the resulting Normal distribution
can be found using the appropriate rules for means and variance
Go over #13-16 in class
13. Suppose  X = 5 and Y = 10. According to the rules for means, what is  X Y ?
14. Suppose  X = 2. According to the rules for means, what is 3 4 X ?
15. Suppose  X2 = 2 and  Y2 = 3 and X and Y are independent random variables.
According to the rules for variances, what is  X2 Y ? What is  X Y ?
16. Suppose  X2 = 4. According to the rules for variances, what is  32 4 X ? What is
 3 4 X ?
6.3
Binomial and Geometric Random Variables (pp.382-401)
1. What is a binomial setting? A binomial setting arises when we perform several
independent trials of the same chance process and record the number of times that a
particular outcome occurs.
2. Describe the conditions of a binomial setting.The four conditions for a binomial
setting are: Binary? The possible outcomes of each trial can be classified as “success” or
“failure.” Independent? Trials must be independent; that is, knowing the result of one
trial must not have any effect on the result of any other trial. Number? The number of
trials n of the chance process must be fixed in advance. Success? On each trial, the
probability p of success must be the same.
3. What is a binomial random variable and what are its possible values? The count X of
successes in a binomial setting is a binomial random variable.
The Practice of Statistics (4th Edition) - Starnes, Yates, Moore
4. Define the parameters of a binomial distribution. The probability distribution of X is a
binomial distribution with parameters n and p, where n is the number of trials of the
chance process and p is the probability of a success on any one trial. The possible values
of X are the whole numbers from 0 to n.
5. Explain the meaning of the binomial coefficient and state the formula.
The number of ways of arranging k successes among n observations is given by the
binomial coefficient
6. Explain how to calculate binomial probabilities.
If X has the binomial distribution with n trials and probability p of success on each
trial, the possible values of X are 0, 1, 2,…, n. If k is any one of these values,
With our formula in hand, we can now calculate any binomial probability
7. What commands on the calculator are used to calculate binomial probabilities?
binompdf(n,p,k) computes P(X = k)
(2ND/VARS) binompdf(n,p,k)
binomcdf(n,p,k) computes P(X ≤ k)
(2ND/VARS) binomcdf(n,p,k)
8. Explain how to calculate the mean and standard deviation of a binomial random
variable.
If a count X has the binomial distribution with number of trials n and probability of
success p, the mean and standard deviation of X are
9. When can the binomial distribution be used to sample without replacement? Explain
why this is an issue. (Discuss in class)
10. What is a geometric setting?
A geometric setting arises when we perform independent trials of the same chance
process and record the number of trials until a particular outcome occurs.
11. Describe the conditions for a geometric setting.
The four conditions for a geometric setting are
Binary? The possible outcomes of each trial can be classified as “success” or “failure.”
Independent? Trials must be independent; that is, knowing the result of one trial must not
have any effect on the result of any other trial.
Trials? The goal is to count the number of trials until the first success occurs.
Success? On each trial, the probability p of success must be the same.
The Practice of Statistics (4th Edition) - Starnes, Yates, Moore
12. What is a geometric random variable and what are its possible values?
The number of trials Y that it takes to get a success in a geometric setting is a
geometric random variable. The possible values of Y are 1, 2, 3,….
13. Describe the parameters of a geometric distribution.
The probability distribution of Y is a geometric distribution with parameter p, the
probability of a success on any trial.
14. What is the formula for geometric probability?
If X has the geometric distribution with probability of success p, the possible values
of Y are the positive integers 1, 2, 3,…. The geometric probability that Y takes
any value is P(Y = k) = (1 − p)k − 1p
15. How is the mean of a geometric random variable calculated?
The mean (expected value) of a geometric random variable X is 1/p.