This document defines key concepts related to random variables including: - A random variable is a numerical measure of outcomes from a random phenomenon. - Probability distributions describe the probabilities associated with random variables. - Expected value refers to the mean or weighted average of a probability distribution. - As the number of trials increases, the actual mean approaches the true mean due to the Law of Large Numbers. - Binomial and geometric distributions model situations with success/failure outcomes and independence between trials.

1. Random Variables
VOCABULARY
RANDOM VARIABLE
PROBABILITY DISTRIBUTION
EXPECTED VALUE
LAW OF LARGE NUMBERS
BINOMIAL DISTRIBUTION
BINOMIAL RANDOM VARIABLE
BINOMIAL COEFFICIENT
GEOMETRIC RANDOM VARIABLE
GEOMETRIC DISTRIBUTION
SIMULATION

2. Key Points
 A random variable is a numerical measure(face up number of a
die) of the outcomes of a random phenomenon(rolling a die)
 If X is a random variable and a and b are fixed numbers, then
μₐ₊ᵦₓ= a+βµₓ and Ợ²ₐ₊ᵦₓ=b²Ợ²x
 If X and Y are random variables, then μₓ₊ᵧ= μₓ + μᵧ
 If X and Y are independent random variables, then Ợ² ₓ₊ᵧ=
Ợ²ₓ + Ợ²ᵧ and Ợ² ₓ₋ᵧ= Ợ²ₓ + Ợ²ᵧ
 As the number of trials in a binomial distribution gets
larger, the binomial distribution gets closer to a normal
distribution

3. Random Phenomenom
Picking a student at random

4. Random Phenomenom
Clicking a Facebook profile at random

5. Random Variable
 A ______ ______ is a numerical measure of the
outcomes of a random phenomenon
 The driving force behind many decisions in
science, business, and every day life is the
question, “What are the chances?”
 Picking a student at random is a random
phenomenon.
 The students grades, height, etc are random
variables that describe properties of the student.

6. Random Variable
The random variables can be: goals inside, goals outside, goals with
right foot, etc..

7. Random Variable
The random variables can be: # of friends, # of miles ran, # of books
recently read, etc

8. Random Variable
The random variables can be categorical as well( top album, movies
watched, favorite artists, etc)

9. Random Variable- Probability distribution
 A _______ ________ is a listing or graphing of
the probabilities associated with a random variable

10. Random Variable- Probability(or population)
distribution
The probability distribution can be used to answer
questions about the variable x( which in this case is the
number of tails obtained when a fair coin is tossed three
times)
Example: What is probability that there is at least one tails
in three tosses of the coin? This question is written as
P(X≥1)
P(X≥1)= P(X=1) + P(X=2)+ P(X=3)= 1/8 +3/8+3/8= 7/8

11. Random variable- discrete and continuous
 _______ random variables takes a countable
number of values(# of votes a certain candidate
receives)
 _______ random variables can take all the possible
values in a given range(the weight of animals in a
certain regions)

12. Discrete Probability Distribution
Probabilities of certain number of surf boards being sold
Doesn’t make sense for someone to purchase 1.3 surfboards

13. Continuous Probability Distribution
Infinite values of x are represented with a Continuous Probability
Distribution

14. Random variable- expected value
 The mean of the probability distribution is referred
to as the ______ ______, and is represented by
μₓ.
which just means that the mean(or expected value)
of a random variable is a weighted average

15. Random Variable- Expected Value
For this probability distribution, the
expected value is
= 0(1/8) + 1(3/8) + 2(3/8) + 3(1/8)= 12/8=
1.5

16. Law of Large Numbers
 The _______ of _______ _______states that the
actual mean of many trials approaches the true mean
of the distribution as the number of trials increases

17. Rules for Means and Variances of Random
Variables

18. Binomial Distribution
 ________ ________ models situations with the
following conditions:
1. Each observation falls into one of just two categories(
success or failure)
2. The number of observations is the fixed number n
3. The n observations are all independent
4. The probability of success, p, is the same for each
observation

19. Binomial Distribution
 For data produced with the binomial model, the
binomial random variable is the number of
successes, X.
 The probability distribution of X is a binomial
distribution
 When finding binomial probabilities, remember that
you are finding the probability of obtaining k successes
in n trials

20. Binomial Distribution
Binomial Coefficient

21. Binomial Distribution
Binomial Coefficient

22. Binomial Distribution- Calculating Binomial
Probability

23. Binomial Distribution- Calculating binomial
probability

24. Mean and Standard deviation of Binomial
Distribution

25. Geometric Distribution
 Each observation falls into one of two categories,
success or failure
 The variable of interest (usually X) is the number of
trials required to obtain the first success
 The n observations are all independent
 The probability of success, p, is the same for each
observation

26. Geometric Distribution
Example: If one planned to roll a die until they got a 5, the random
variable X= the number of trials until the first 5 occurs.
Find the probability that it would take 8 rolls given that all the
conditions of the geometric model are met

27. Geometric Distribution
Expected Value of Geometric Distributions
If X is a geometric random variable with probability of success P
on each trial, then the mean or _______ _______ of the
random variable is μ= 1/p.