Week 4 - Probability Distributions

Question 1

What is a random variable?

Answer 1

A random variable is a function that associates a real number with each element in the sample space.

Question 2

What is a discrete sample space?

Answer 2

If a sample space contains a finite number of possibilities or an unending sequence
with as many elements as there are whole numbers, it is called a discrete sample
space.

Question 3

What is a continious sample space?

Answer 3

If a sample space contains an infinite number of possibilities equal to the number of
points on a line segment, it is called a continuous sample space.

Question 4

What is a discrete random variable?

Answer 4

A random variable is called a discrete random variable if its set of possible outcomes is countable.

Question 5

What is a finite random variable?

Answer 5

X is a finite random variable if the range is a finite set.

Question 6

What does a probability distribution do?

Answer 6

A probability distribution for a random variable summarizes or models the probabilities associated with the events for that random variable.
The distribution takes a different form depending on whether the random variable is discrete or continuous.

Discrete Probability Distribution
A listing of all possible distinct (elementary) events for a random variable and their probabilities of occurrence.

Question 7

midterm question

What is binomial probability distributions used for?

Answer 7

Binomial is used for random variables that have only two mutually exclusive events.

continued in the footnote the whole answer.

• The random variable is for a sample that consists of a fixed number of experimental trials.
• The random variable has only two mutually exclusive and collectively exhaustive events, typically labeled as success and failure.
• The probability of an event being classified as a success, p, and the probability of an event being classified as a failure, 1 – p, are both constant in all experimental trials.
• The event (success or failure) of any single experimental trial is independent of (not influenced by) the event of any other trial.

Question 8

possible mid makeup question

What is Poisson probability distribution used for?

Answer 8

Poisson, is used when you are counting the number of outcomes that occur in a unit.

Examples:
Number of cars passing from street per hour,
Number of customers arriving to a bank between 12 noon to 1 p.m.
Number of computer failures per day,
Number of failures per meter square of a floor cover.

more in footnote

In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time and/or space if these events occur with a known average rate and independently of the time since the last event.
The number of events, with same probability and independent from each other is expressed with Poisson distribution.

Question 9

What are Binomial Distributions Characteristics?

Answer 9

Mean ( n x p ),
Variance ( n x p x (1-p) ),
Standard Deviation ( sqrt(var) )

Question 10

mid question

Examples for binomial distribution?

Answer 10

Number of heads in 10 coin tosses.
Pass/fail outcomes for a group of students on a test.
Defective/non-defective items in a production batch.
Votes for a candidate in a series of elections.
Patients responding positively/negatively to a treatment.