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Unit 4: Probability Distribution✏️ - Coggle Diagram
Unit 4: Probability Distribution
✏️
Random Variables
Combination
To add or subtract independent random variables, simply do so (mean)
Calculate the standard deviation of the sum/difference of two or more independent random variables with formula below
(don't subtract variance!)
:
Transformation
To multiply independent random variables by a constant, simply do so (mean)
Calculate the transformed standard deviation by multiplying standard deviation of random variable by constant (do NOT add)
z-scores are not affected by transformations!
Discrete Probability Distribution
Probability Distribution: Can be represented in a table, graph, or function. It shows the probabilities associated with the values of the random variable. All probabilities must sum up to one.
Discrete Random Variable: A variable that is counted
Mean:
p = probabilities of x-values
Standard Deviation:
Interpretation: The average (context) is (mean) and will typically be off by (standard deviation & context).
Binomial Probabilities
What qualifies as a binomial random variable?
B: two possible outcomes
I: trials are independent
N: number of trials is fixed
S: probability of success remains constant from one trial to the other
Calculating binomial probabilities
binompdf: when the x-value is exactly 1 number
binomcdf: when the x-value is a range, indicated by "at least," "more than," etc. (on Ti-84, the inserted x-value is inclusive)
Mean:
Standard Deviation:
Geometric Probabilities
What qualifies as a geometric random variable?
B: two possible outcomes
I: trials are independent
T: number of trials are NOT fixed
S: probability of success remains constant from one trial to the other
Calculating geometric probabilities
Mean:
Standard Deviation:
A geometric random variable occurs when you repeat an independent event over and over until you get your first success
Probability Strategies
Vocabulary
Probability: Describes the proportion of times outcomes would occur in many repetitions
Outcome: An outcome is a result of a trial of a random process
Event: An event is a collection of outcomes
Law of Large Numbers: States that simulated probabilities get closer to the true probability as the number of trial increases
Complement of an Event: The probability of the complement of an event is equal to 1 minus the probability of an event occurring
Sample Space: Is the set of all possible outcomes (non-overlapping) of a random process
Mutually Exclusive Events: When two events cannot happen at the same time (e.g. rolling a die and landing on 3, rolling a die and landing on 4)
1. Simulation
testing the probability of a sample occurring by change
requires lots of repetition
possible methods: RNG, rolling dice, flipping coin, drawing a paper out of all that are equally-sized
(1) state question of interest (2) plan how to simulate chance process (3) perform many repetitions of simulation (4) conclude using
results in context
2. Venn Diagram
usually compares two events
Union: addition
And: multiplication
3. Two-Way Table
two or more events; joint probabilities
when two or more probabilities are unknown
4. Tree Diagram
sequential events with lots of percentages
5. Formula Sheet!
Conditional Probability: the probability that an event will occur "given that" or "if" another event takes place
Independent Events: the probability of independent events happening simultaneously can be calculated by multiplying their probabilities together
Independent events are NOT the same as mutually exclusive events! Independent events can still happen at the same time, yet they do not affect the probability of each other happening
Simplify the formula for conditional probability to test if two events are independent:
