### 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:

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