Chapter 7: Random Variables and Discrete Probability Distributions

Question 1

Random variable

Answer 1

A function or a rule that assigns a number to each outcome of an experiment

Name of random variable = uppercase letter

Value of random variable = lowercase letter

Probability X has the value of x = P(X= x) or just P(x)

Question 2

Discrete random variable

Answer 2

Random variable that can take on a countable number of values

Question 3

Continuous random variable

Answer 3

Random variable whose values are uncountable (could be anything - infinitely divisible)

Question 4

Probability distribution

Answer 4

A table, formula, or graph that describes the values of a random variable and the probability associated with these values

When multiple outcomes are possible to reach a single value then the sum of these possible outcomes is the probability of that value

Probability distribution of a sample represents the probability distribution of the population

Question 5

Requirements for a distribution of a discrete random variable

Answer 5

1) probability of x is between 0 and 1 for all values of x

2) some of the probability of all possible values of x = 1

Question 6

Population mean

Answer 6

Aka expected value (of the random variable) E(X)

Weighted average of all values

Weight = probability

E(X) = sum of all individual values of x multiplied by their probability

Question 7

Population variance

Answer 7

The weighted average of all the squared deviations from the mean

So sum of all difference between value of x and mean squared times the probability of that value of x

Shortcut:
Sum of all (values of x squared multiplied by the probability of x) less the mean squared

Question 8

Laws of expected value

Answer 8

Where X is the random variable and c is a constant

E(c) = c
E(X+c) = E(X)+c
E(cX) = cE(X)

Question 9

Laws of variance

Answer 9

Where X is the random variable and c is a constant

V(c)=0
V(X+c)= V(X)
V(cX) =c^2V(X)

Question 10

Binomial experiment

Answer 10

• consists of a fixed number of trials (number of trials = n)
• each trial has two possible outcomes, one labeled success other labeled failure
• probability of success is p. Probability of failure is 1-p
• trials are independent (outcome of one trial does not affect the outcome of others)

Question 11

Bernoulli Process

Answer 11

• each trial has two possible outcomes, one labeled success other labeled failure
• probability of success is p. Probability of failure is 1-p
• trials are independent (outcome of one trial does not affect the outcome of others)

Binomial experiment without fixed number of trials

Question 12

Binomial random variable

Answer 12

Random variable is the number of successes in the n trials

A discrete variable (only integers)

Question 13

Binomial probability distribution

Answer 13

The probability of x successes in a binomial experiment with n trials (where the probability of success = p)

P(x)= (n!/ x!(n-x)!)p^x(1-p)^(n-x)

Remember 0!=1

(First half determines number of ways to get that number of successes second half determines probability of each way)

Question 14

Cumulative probability

Answer 14

The probability that a random variable is less than or equal to a value

For value x finding P(X<=x)

Sum of probabilities of all random variable values x and below

Question 15

Binomial table

Answer 15

Provides cumulative binomial probabilities for given values of n (tests) and p (probability of success in a single experiment)

Question 16

Using binomial table to find the binomial probability of P(X>=x)

Answer 16

Basically using the compliment rule. The probability that X is greater than or equal to x value is 1 minus the probability it is less than or equal to x-1 so:

P(X>=x) = 1 - P(X<=[x-1])

Question 17

Using the binomial table to find the binomial probability or P(X=x)

Answer 17

P(X=x) = P(X<=x) - P(X<=[x-1])

Question 18

Binomial probability in excel

Answer 18

=BINOMDIST([x], [n], [p], True or False)

x = number of successes
n = total number of tests 
p = probability of success
True = cumulative probability (P(X<=x)
False = probability of individual value x

Question 19

Mean of a binomial random variable

Answer 19

= n * p

Number of tests * probability of success

Question 20

Variance of a binomial random variable

Answer 20

= np(1-p)

Number of tests * probability of success * probability of failure

Question 21

Standard distribution of a binomial random variable

Answer 21

Square root of variance

=√np(1-p)

Question 22

Poisson random variable

Answer 22

The number of successes that occur in a period of time or an interval of space in a Poisson experiment

Question 23

Poisson experiment

Answer 23

Characteristics:

• number of success that occur in any interval is independent of the number of successes that occur in any other interval
• the probability of a success in an interval is the same for all equal sized intervals
• the probability of a success in an interval is proportional to the size of the interval
• the probability of more than one success in an interval approaches 0 as the interval becomes smaller

Question 24

Poisson probability distribution

Answer 24

A discrete probability distribution

probability of x successes in a specific interval = e to the negative power of the mean number of successes times the mean number of successes to the power of x all divided by x factorial

P(x) = ((e^- mean)*(mean^x))/x!

e= base of the natural logarithm

Question 25

Variance of a Poisson random variable

Answer 25

Equal to the mean of the random variable

Question 26

Calculating the Poisson probability distribution when the interval does not match

Answer 26

• find the proportion of the interval of interest to the given mean interval
• multiply or divide the mean by the relative proportion
• solve qsbusyal

Question 27

Cumulative Poisson probabilities by formula

Answer 27

Find each of the discrete probability and sun

Question 28

Poisson table

Answer 28

Shows cumulative Poisson probabilities for selected mean values

Gives values for P(x)<=x

Question 29

Using Poisson table to find Poisson probability P(X>=x)

Answer 29

P(X>=x)= 1 - P(X<=[x-1])

Question 30

Using Poisson probability table to find an individual value of x

Answer 30

P(X=x) = P(X<=x) - P(X<=[x-1])

Question 31

Poisson probability in excel

Answer 31

=POISSON([x],[mean], True or False

True = cumulative probability, false = individual