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 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.

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.

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 statsprint(stats.binom.pmf(4, 10, 0.5))# Output:# 0.20507812500000022

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.

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 statsprint(stats.binom.cdf(4, 10, 0.5))# Output:# 0.3769531250000001

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.

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.

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 *union* of two sets encompasses any element that exists in either one or both of them. We can represent this visually as a *venn diagram* as shown. Union is often represented as:

`$(A\ or\ B)$`

The intersection between two sets encompasses any element that exists in BOTH sets and is often written out as:

`$(A\ and\ B)$`

If there are two events, A and B, the addition rule states that the probability of event A or B occurring is the sum of the probability of each event minus the probability of the intersection:

`$P(A\ or\ B) = P(A) + P(B) - P(A\ and\ B)$`

If the events are mutually exclusive, this formula simplifies to:

`$P(A\ or\ B) = P(A) + P(B)$`

The multiplication rule is used to find the probability of two events, *A* and *B*, happening simultaneously. The general formula is:

`$P(A \text{ and } B) = P(A) \cdot P(B \mid A)$`

For independent events, this formula simplifies to:

`$P(A \text{ and } B) = P(A) \cdot P(B)$`

This is because the following is true for independent events:

`$P(B \mid A) = P(B)$`

The tree diagram shown displays an example of the multiplication rule for independent events.

The complement of a set consists of all possible outcomes outside of the set.

Let’s say set *A* is rolling an odd number with a 6-sided die: *{1, 3, 5}*. The complement of this set would be rolling an even number: *{2, 4, 6}*.

We can write the complement of set *A* as *A ^{C}*. One key feature of complements is that a set and its complement cover the entire sample space. In this die roll example, the set of even numbers and odd numbers would cover all possible rolls:

Two events are *independent* if the occurrence of one event does not affect the probability of the other one occurring.

Let’s say we have a bag of five marbles: three are red and two are blue. If we select two marbles out of the bag WITH replacement, the probability of selecting a blue marble second is independent of the outcome of the first event.

The diagram below outlines the independent nature of these events. Whether a red marble or a blue marble is chosen randomly first, the chance of selecting a blue marble second is always 2 in 5.

Two events are *dependent* if the occurrence of one event does affect the probability of the other one occurring.

Let’s say we have a bag of five marbles: three are red and two are blue. If we select two marbles out of the bag WITHOUT replacement, the probability of selecting a blue marble second depends on the outcome of the first event.

The diagram below outlines this dependency. If a red marble is randomly selected first, the chance of selecting a blue marble second is 2 in 4. Meanwhile, if a blue marble is randomly selected first, the chance of selecting a blue marble second is 1 in 4.

Two events are considered *mutually exclusive* if they cannot occur at the same time. For example, consider a single coin flip: the events “tails” and “heads” are mutually exclusive because we cannot get both tails and heads on a single flip.

We can visualize two mutually exclusive events as a pair of non-overlapping circles. They do not overlap because there is no outcome for one event that is also in the sample space for the other.

Conditional probability is the probability of one event occurring, given that another one has already occurred. We can represent this with the following notation:

```
$\begin{aligned}
\text{Probability of event A occurring given event B has occurred} \\
P(A \mid B) \\
\end{aligned}$
```

For independent events, the following is true for events *A* and *B*:

```
$\begin{aligned}
P(A \mid B) = P(A) \\
\text{and} \\
P(B \mid A) = P(B) \\
\end{aligned}$
```

Bayes’ theorem is a useful tool to find the probability of an event based on prior knowledge. The formula for Bayes’ theorem is:

`$P(B \mid A) = \frac{P(A \mid B) \cdot P(B)}{P(A)}$`

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.

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:

`$X \sim Binomial(n, p), \; E(X) = n \times p$`

`$Y \sim Poisson(\lambda), \; E(Y) = \lambda$`

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:

`$X \sim Binomial(n, p), \; Var(X) = n \times p \times (1-p)$`

`$Y \sim Poisson(\lambda), \; Var(Y) = \lambda$`

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:

`$E(X + Y) = E(X) + E(Y)$`

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:

`$E(X + c) = E(X) + c$`

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:

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

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

Mathematically:

`$Var(X + c) = Var(X)$`

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:

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