Probability Distributions

Probability Distribution

Mathematical function that describes the likelihood of different outcomes in a random process.
Types: Discrete (Binomial, Poisson) and Continuous (Normal, Exponential).
Understanding distributions helps in selecting proper statistical methods.

Common Distributions:

Normal Distribution: Bell-shaped curve, defined by mean and standard deviation.
Bernouli Distribution: Models a single trial with two possible outcomes: success (1) and failure (0).
Binomial Distribution: Models number of successes in a fixed number of independent trials.
Poisson Distribution: Models number of events in a fixed interval of time.

Models single trial with two possible outcomes: success(1) or failure(0).
Parameter p represents the probability of success.
Scenarios: Coin flip p=0.5, Customer purchase p=0.2, Email open p=0.3.

Number of successes in a fixed number of independent Bernoulli trials.
Parameter n represents the number of trials, and p represents the probability of success in each trial.
Scenarios: Number of heads in 10 coin flips, number of customers who purchase out of 100 visitors.

Number of events occurring in a fixed interval of time or space.
Parameter λ (lambda) denotes average number of events in the interval.
Used for modeling rare events or counting occurrences in fixed intervals.
Scenarios: Number of customer arrivals at a store in an hour, number of emails received in a day.

Collect data from past.
Probability aren't guessed but estimated from real world frequencies.
Formulae: For p: Success / Total Trial. For λ (Lambda): Average number of events per interval. For Mean and Sigma: Calculate from past data.

If you are running a cake shop, Distribution selection guide:

Distribution	Use Case	Example
Normal	Continuous data, symmetric	Average daily sales amount.
Bernoulli	Single binary outcome	Probability of a customer buying a cake.
Binomial	Fixed number of independent trials	Number of customers who buy a cake out of 100 visitors.
Poisson	Count data, rare events	Number of customers arriving at the shop per hour.

Distribution selection guide based on use case and example scenarios for a cake shop

Key Functions for Probability Distributions:

Function	Description	Applicable To
PMF (Probability Mass Function)	Gives probability of each outcome for discrete distributions.	Discrete Distributions (e.g., Bernoulli, Binomial, Poisson)
PDF (Probability Density Function)	For continuous distributions, describes relative likelihood of outcomes.	Continuous Distributions (e.g., Normal, Exponential)
CDF (Cumulative Distribution Function)	Gives probability that a random variable is less than or equal to a certain value.	All Distributions

Range of values that likely contain the true population parameter.
Calculated from sample data and provides a measure of uncertainty.
If true population mean is fish in a lake: - Point Estimate: We throw single spear at fish. Likely miss. - Confidence Interval: We throw a net around fish. Indicate how often net catches fish.
To calculate 95% CI for a Normal Distribution. We need: - Sample mean (X̄) - Standard Error (σ/√n) - Z-score for 95% confidence (1.96) CI = X̄ ± Z * SE