VOCABULARYRANDOM VARIABLE
PROBABILITY DISTRIBUTIONEXPECTED VALUE
LAW OF LARGE NUMBERSBINOMIAL DISTRIBUTION
BINOMIAL RANDOM VARIABLEBINOMIAL COEFFICIENT
GEOMETRIC RANDOM VARIABLEGEOMETRIC DISTRIBUTION
SIMULATION
Random Variables
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 + μᵧIf 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
Random Phenomenom
Picking a student at random
Random Phenomenom
Clicking a Facebook profile at random
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.
Random Variable
The random variables can be: goals inside, goals outside, goals with right foot, etc..
Random Variable
The random variables can be: # of friends, # of miles ran, # of books recently read, etc
Random Variable
The random variables can be categorical as well( top album, movies watched, favorite artists, etc)
Random Variable- Probability distribution
A _______ ________ is a listing or graphing of the probabilities associated with a random variable
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
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)
Discrete Probability Distribution
Probabilities of certain number of surf boards being sold
Doesn’t make sense for someone to purchase 1.3 surfboards
Continuous Probability Distribution
Infinite values of x are represented with a Continuous Probability Distribution
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
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
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
Rules for Means and Variances of Random Variables
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 independent4. The probability of success, p, is the same for
each observation
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
Binomial Distribution
Binomial Coefficient
Binomial Distribution
Binomial Coefficient
Binomial Distribution- Calculating Binomial Probability
Binomial Distribution- Calculating binomial probability
Mean and Standard deviation of Binomial Distribution
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 independentThe probability of success, p, is the same for
each observation
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
Geometric Distribution
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.
Expected Value of Geometric Distributions
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