# Probability Distributions

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### Random Variables

Random variables are functions with numerical outcomes that occur with some level of uncertainty. For example, rolling a 6-sided die could be considered a random variable with possible outcomes {1,2,3,4,5,6}.

### Discrete and Continuous Random Variables

Discrete random variables have countable values, such as the outcome of a 6-sided die roll.

Continuous random variables have an uncountable amount of possible values and are typically measurements, such as the height of a randomly chosen person or the temperature on a randomly chosen day.

### Probability Mass Functions

A probability mass function (PMF) defines the probability that a discrete random variable is equal to an exact value.

In the provided graph, the height of each bar represents the probability of observing a particular number of heads (the numbers on the x-axis) in 10 fair coin flips.

### Probability Mass Functions in Python

The binom.pmf() method from the scipy.stats module can be used to calculate the probability of observing a specific value in a random experiment.

For example, the provided code calculates the probability of observing exactly 4 heads from 10 fair coin flips.

import scipy.stats as stats
print(stats.binom.pmf(4, 10, 0.5))
# Output:# 0.20507812500000022

### Cumulative Distribution Function

A cumulative distribution function (CDF) for a random variable is defined as the probability that the random variable is less than or equal to a specific value.

In the provided GIF, we can see that as x increases, the height of the CDF is equal to the total height of equal or smaller values from the PMF.

### Calculating Probability Using the CDF

The binom.cdf() method from the scipy.stats module can be used to calculate the probability of observing a specific value or less using the cumulative density function.

The given code calculates the probability of observing 4 or fewer heads from 10 fair coin flips.

import scipy.stats as stats
print(stats.binom.cdf(4, 10, 0.5))
# Output:# 0.3769531250000001

### Probability Density Functions

For a continuous random variable, the probability density function (PDF) is defined such that the area underneath the PDF curve in a given range is equal to the probability of the random variable equalling a value in that range.

The provided gif shows how we can visualize the area under the curve between two values.

### Probability Density Function at a Single Point

The probability that a continuous random variable equals any exact value is zero. This is because the area underneath the PDF for a single point is zero.

In the provided gif, as the endpoints on the x-axis get closer together, the area under the curve decreases. When we try to take the area of a single point, we get 0.

### Parameters of Probability Distributions

Probability distributions have parameters that control the exact shape of the distribution.

For example, the binomial probability distribution describes a random variable that represents the number of sucesses in a number of trials (n) with some fixed probability of success in each trial (p). The parameters of the binomial distribution are therefore n and p. For example, the number of heads observed in 10 flips of a fair coin follows a binomial distribution with n=10 and p=0.5.

### The Poisson Distribution

The Poisson distribution is a probability distribution that represents the number of times an event occurs in a fixed time and/or space interval and is defined by parameter λ (lambda).

Examples of events that can be described by the Poisson distribution include the number of bikes crossing an intersection in a specific hour and the number of meteors seen in a minute of a meteor shower.

### Expected Value

The expected value of a probability distribution is the weighted (by probability) average of all possible outcomes. For different random variables, we can generally derive a formula for the expected value based on the parameters.

For example, the expected value of the binomial distribution is n*p.

The expected value of the Poisson distribution is the parameter λ (lambda).

Mathematically:

X \sim Binomial(n, p), \; E(X) = n \times p
Y \sim Poisson(\lambda), \; E(Y) = \lambda

### Variance of a Probability Distribution

The variance of a probability distribution measures the spread of possible values. Similarly to expected value, we can generally write an equation for the variance of a particular distribution as a function of the parameters.

For example:

X \sim Binomial(n, p), \; Var(X) = n \times p \times (1-p)
Y \sim Poisson(\lambda), \; Var(Y) = \lambda

### Sum of Expected Values

For two random variables, X and Y, the expected value of the sum of X and Y is equal to the sum of the expected values.

Mathematically:

E(X + Y) = E(X) + E(Y)

### Adding a Constant to an Expected Value

If we add a constant c to a random variable X, the expected value of X + c is equal to the original expected value of X plus c.

Mathematically:

E(X + c) = E(X) + c

### Multiplying an Expectation by a Constant

If we multiply a random variable X by a constant c, the expected value of c*X equals the original expected value of X times c.

Mathematically:

E(c \times X) = c \times E(X)

### Adding a Constant to Variance

If we add a constant c to a random variable X, the variance of the random variable will not change.

Mathematically:

Var(X + c) = Var(X)

### Multiplying Variance by a Constant

If we multiply a random variable X by a constant c, the variance of c*X equals the original expected value of X times c squared.

Mathematically:

Var(c\times X) = c^2 \times Var(X)