Statistical Distributions

Question 1

What is a discrete random variable?

Answer 1

A variable that can take a certain number of different values with different probabilities.

Question 2

What is a probability distribution?

Answer 2

A table showing all the possible values a discrete random variable can take and the probability that it takes them.

Question 3

How is a discrete random variable represented?

Answer 3

Usually as an upper case letter such as X.

Question 4

How are the particular values of X represented?

Answer 4

With a lower case version of the letter such as x.

Question 5

What is a probability function?

Answer 5

A formula that generates the probability of X taking the value x or every possible x.

Question 6

How is a probability function written?

Answer 6

P(X=x)

Question 7

What is the sum of all possible values that a discrete random variable can take?

Answer 7

1

Question 8

What is a uniform distribution?

Answer 8

A random variable where every possible value of X is equally likely.

Question 9

What is the cumulative distribution function?

Answer 9

A function that gives the probability that X will be less than or equal to a particular value of x.

Question 10

How is the cumulative distribution function written?

Answer 10

F(x) or P(X <= x)

Question 11

How do you find F(x) for a given value of x?

Answer 11

Add up all the probabilities of X less than or equal to x.

Question 12

How do you work out the number of ways that n objects where x are identical can be arranged?

Answer 12

n!/x!

Question 13

How do you work out the number of ways that n objects can be arranged where there are only two types of objects; x of one type and (n-x) of the other?

Answer 13

n!/(x!(n - x)!)

Question 14

What is the binomial coefficient?

Answer 14

n!/(x!(n - x)!) or nCr

Question 15

What do you use the binomial coefficient to work out?

Answer 15

The number of ways to arrange x successes and (n - x) failures for a trial that can result in either success of failure.

Question 16

What are the five conditions that need to be satisfied for a random variable X to follow a binomial distribution?

Answer 16

1. There is a fixed number (n) of trials.
2. Each trial involves either ‘success’ or ‘failure’.
3. All the trials are independent.
4. The probability of ‘success’ (p) is the same in each trial.
5. The variable is the total number of successes in the n trials.

Question 17

How do you write that X follows a binomial distribution?

Answer 17

X~B(n, p), where n is the number of trials and p is the probability of success.

Question 18

What is the probability function of X if it follows a binomial distribution?

Answer 18

For X~B(n, p):

P(X = x) = nCx * p^x * (1 - p)^(n - x) for x = 0, 1, 2, …, n.

Question 19

What is the mean expected value of a random variable?

Answer 19

The value you’d expect x to take on average: If X~B(n, p), then Mean or expected value E(X) = n * p