Module 5 Flashcards
What do you need to calculate probability?
The number of trials (n) and the number of outcomes per trial (k).
What is the formula for multistep probability?
Multiply the number of outcomes per trial for all trials. Example: For 3 coins, (2)(2)(2)=8.
What is 5!?
5!=5×4×3×2×1=120.
What is a discrete uniform distribution?
A distribution where all outcomes are equally likely, like rolling a die.
How do you validate a probability distribution?
- Probabilities must be between 0 and 1.
- The sum of probabilities must equal 1.
What is the probability of rolling two 6’s with two dice?
Total outcomes = 6×6=36.
Probability of two 6’s = 1/36
.
How many possible outcomes are there when rolling 5 dice?
(6)(6)(6)(6)(6) = 6^5 =7776
What are the conditions for a discrete probability distribution?
0<P(X=x)<1 for all outcomes.
∑P(X=x)=1.
What is a random variable?
A variable that takes on numerical values based on the outcomes of a random event
What is a probability mass function (PMF)?
A function that assigns probabilities to each value of a discrete random variable.
What is the relationship between trials and outcomes in probability?
Multiply the number of outcomes for each trial to find total outcomes.
What are the four types of probability distributions?
Discrete distributions
Binomial
Poisson
Continuous distributions
Normal
Exponential
What are some examples of a binomial distribution?
A customer defaults or does not default on a loan.
A consumer reacts positively or negatively to a social media campaign.
A drug is either effective or ineffective.
What are the components of a binomial distribution?
n: Number of trials.
x: Number of successes.
P: Probability of success.
1−P: Probability of failure.
What is the formula for a binomial probability?
P(X=x)= (n x) P^x (1– P)^n-x
(n x) = number of ways to choose x successes out of n trials
P^z = probability of x successes
(1 – P)^n-x = probability of n–x failures
What are the properties of a binomial experiment?
Fixed number of trials (n).
Each trial is independent.
Only two possible outcomes (success/failure).
The probability of success (P) is the same for each trial.
What is the difference between discrete and continuous probability distributions?
Discrete: Outcomes are countable (e.g., Binomial, Poisson).
Continuous: Outcomes are uncountable and measured over a range (e.g., Normal, Exponential).
Binomial Distribution
Discrete probability distribution for a fixed number of independent trials, each with two possible outcomes.
Poisson Distribution
Describes the number of successes in a fixed time or space.
Examples:
Spam emails in a month.
Cars in line at McDonald’s in an hour.
Expected Value
The weighted average of all possible values for a random variable.
Variance
Measures spread around the mean.
Risk Preferences
Risk-Averse: Avoid risk without positive gain.
Risk-Neutral: Focus on expected gain only.
Risk-Loving: Willing to accept risk, even with negative gain.
Definition of the Poisson Distribution
Used for counting the number of occurrences (successes) of an event over a specific interval of time or space.
A Poisson process satisfies these conditions:
The number of successes in an interval can be any integer from 0 to infinity.
Successes in non-overlapping intervals are independent.
The probability of success is proportional to the size of the interval.
Excel Formula for Poisson Distribution
Syntax: POISSON.DIST(x, mean, cumulative)
x: The number of occurrences (successes).
mean: The expected value (average rate of occurrence).
Cumulative:
-TRUE: Returns cumulative probability P(X ≤ x).
-FALSE: Returns probability mass P(X = x).
Example: Mean = 15, Find P(X=10)
Use the formula: POISSON.DIST(10, 15, FALSE).
Key Properties of Poisson Random Variable
Discrete probability distribution.
Expected value (mean): λ
Variance: λ.
Binomial vs. Poisson
Binomial Probability: P(6≤X≤11)=P(X≤11)−P(X≤5).
Used for finite trials with success probability (e.g., graduate degree example).
Poisson:
Used for events occurring over time or space (e.g., foreclosures).
