Chapter 4 - Discrete Distributions

Question 1

Random variable

Answer 1

• random variable = function that assigns numeric values to different events in a sample space
• two types of random variables = discrete, continuous

Question 2

Discrete random variable

Answer 2

a random variable for which there exists a discrete (finite) set of numeric values

Question 3

Continuous random variable

Answer 3

a random variable whose possible values cannot be enumerated (infinite)

Question 4

Probability-mass function

Answer 4

• the values taken by a discrete random variable and its associated probabilities can be expressed by a rule or relationship called a probability-mass function
• assigns to any possible value r a discrete random variable X, the probability P(X = r)
• this assignment is made for all values r that have positive probability

Question 5

Expression of a probability-mass function

Answer 5

• a pmf can be displayed in a tabular form, or it can be expressed as a mathematical formula giving the probabilities of all possible values
• the probability of any particular value must be between 0 and 1, and the sum of the probabilities of all values must be exactly equal to 1

Question 6

Frequency distributions

Answer 6

• a list of each value in the data set and a corresponding count of how frequently the value occurs
• the frequency distribution can be considered as a sample analog to a probability distribution
• frequency distribution gives the actual proportion of points in a sample that correspond to specific values

Question 7

Goodness-of-fit

Answer 7

the appropriateness of a model can be assessed by comparing the observed sample-frequency distribution with the probability distribution

Question 8

Expected value of a discrete random variable

Answer 8

• if a random variable has a large number of values with positive probability, then the pmf is not a useful summary
• measures of location and spread can be developed for a random variable in the same way as for samples
• expected value is also called population mean

Question 9

Variance of a discrete random variable

Answer 9

• the analog of the sample variance for a random variable
• also called population variance
• the variance represents the spread, relative to the expected value, of all values that have positive probability
• approximately 95% of the probability mass falls within two standard deviations of the mean of a random variable

Question 10

Cumulative-distribution function

Answer 10

• for a discrete random variable, the cdf looks like a series of steps, called the step function
• with the increase in number of values, the cdf approaches that of a smooth curve

Question 11

Permutations

Answer 11

• in a matched-pair design, each sample/case is matched with a normal control of the same sex and age
• once the first control is chosen, the second control can be chosen in (n-1) ways

Question 12

Combinations

Answer 12

• in an unmatched study design, cases and controls are selected in no particular order
• thus, the method of selecting n things taken k at a time without respect to order is referred to as the number of combinations

Question 13

Binomial distribution

Answer 13

• a sample of n independent trials, each of which can have only two possible outcomes
• the probability of a success at each trial is assumed to be some constant p
• the probability at each trial is 1-p=q
• number of trials n is finite, and the number of events can be no larger than n

Question 14

Calculating binomial probabilities

Answer 14

• for sufficiently large n, the normal distribution can be used to approximate the binomial distribution and tables of the normal distribution can be used to evaluate binomial probabilities
• if the sample size is not large enough to use normal approximation, then an electronic table can be used

Question 15

Expected value and variance of the binomial distribution

Answer 15

• the expected number of successes in n trials is the probability of success in one trial multiplied by p, which equals np
• for a given number of trials n, the binomial distribution has the highest variance when p=1/2
• variance decreases as p moves away from ½ becoming 0 when p=0 or p=1

Question 16

Poisson distribution

Answer 16

• the Poisson distribution is the second most frequently used discrete distribution after binomial distribution
• it is usually associated with rare events
• the number of trials is essentially infinite and the number of events can be indefinitely large
• however, probability of k events becomes very small as k increases
• the plotted distribution tends to become more symmetric as the time interval increases, or more specifically, as u increases
• for a Poisson distribution with parameter u, the mean and variance are both equal to u

Question 17

Poisson approximation to the binomial distribution

Answer 17

• the binomial distribution with large n and small p can be accurately approximated by a Poisson distribution with parameter u=np
• the mean of this distribution is given by np and the variance by npq