Chapter 5

Question 1

Random variable

Answer 1

A variable whose value is determined by the outcome of a random experiment.
Can take on a countable number of distinct values.
Not infinite = could put it in a table.

Question 2

Continuous random variables

Answer 2

A variable that can be exactly put in a chart.

Question 3

Probability distribution of a random variable

Answer 3

Describes how the probabilities are distributed over all possible values.

Question 4

Mean of a discrete random variable

Answer 4

Represents the long-run average outcome of the random variable over many trials or observations.
u = E(X) = (sum of x’s)(P(x))
This is a weighted average (weights are probabilities)

Question 5

Steps to find the mean/ expected value of a discrete random variable:

Answer 5

1. List the possible values of the discrete random variable
2. Find the probability associated w/ each possible value
3. Multiply each value of the Random variable by its corresponding probability
4. Sum all the products to get the mean/expected value.

Question 6

Variance of a discrete random variable

Answer 6

Measures the spread of its probability distribution.
o squared = sum of x squared x P(x) - mean squared

Question 7

Standard deviation of a discrete random variable

Answer 7

Take the square root of the variance.

Question 8

3 common discrete random variables

Answer 8

1. binomial
2. Hypergeometric
3. Poisson

Question 9

4 conditions of a binomial experiment

Answer 9

1. There are identical n “trials”
2. Each trial only has 2 possible outcomes trial
3. Probabilities for the 2 outcomes remain constant for each trial
4. The trials are independent

Question 10

Binomial probabilities

Answer 10

Let x be a random variable that counts the number of “successful” trials.
P(x) = (nCx) (p to the power of x) (q to the power of n - x)
n = total number of trials
p = probability of success
q = p - 1
x = number of successes in n trials

Question 11

Table of binomial probabilities

Answer 11

You can use a table to calculate binomial probabilities for common P values.

Question 12

Shape of the binomial distributions

Answer 12

Symmetric if P = 0.5
Right-skewed if p<0.5
Left-skewed if P > 0.5

Question 13

Mean of the binomial distribution

Answer 13

u = np
n = number of trials
p = probability

Question 14

Variance of a discrete random variable

Answer 14

o squared = npq
n = number of trials
p = probability
q = p - 1

Question 15

Hypergeometric distribution

Answer 15

Involves drawing a sample with out replacement from a finite population.

Question 16

Hypergeometric probability

Answer 16

p(x) = (rCx) (N-rCn-x) / (NCn)
N = total number of elements in population
n = number of trials (sample size)
r = number of successes in the population
x = number of successes in the n trials

Question 17

Mean of hypogeometric distribution

Answer 17

u = nr/N

Question 18

Variance of the hypogeometric distribution

Answer 18

o squared = [ nr (N - r) (N - n) ] / [N squared (N - 1)]

Question 19

Poisson distribution

Answer 19

Used in situations where we are interested in the number of times an event occurs within a fixed interval of time (or space), and the events happen indefinitely.

Question 20

3 conditions to apply Poisson distribution

Answer 20

1. X is a discrete random variable
2. The occurrences are random
3. The occurrences are independent

Question 21

Poisson probabilities

Answer 21

P(x) = (lambda to the power of x) (e to the power of negative lambda) / (x factorial)
Lambda = the mean number of occurrences in that interval
e = Euler’s number

Question 22

Table of poisson probabilities

Answer 22

Instead of using the formula to calculate the probabilities, you can use a table for common values of lambda.

Question 23

Mean of Poisson distribution

Answer 23

u = lambda
o squared = lambda
o = lambda square rooted