Definition

Probability Distribution

Definition

Probability Mass Function (PMF)

Theorem:
Properties of a Probability Mass Function
For a discrete random variable $X$ with PMF $P(X=x)$, the following properties must hold:

  • $P(X=x) \geq 0$ for all values of $x$
  • $\displaystyle \sum_{i=1}^n P(X=x) = 1$

Example

Let $X$ be a random variable with the following probability distribution: $$\begin{array}{c|ccccc} x & -2 & -1 & 0 & 1 & 2 \\ \hline f(x) & 0.2 & 0.4 & 0.1 & 0.2 & 0.1\end{array}$$

Example

Let $X$ be a random variable with the following probability distribution:$$f(x)=\frac{2x+1}{25} \quad x=0,1,2,3,4$$

Example

An Al system is monitoring user interactions with a new app feature to evaluate its usability. Each interaction is classified as either ``successful`` (the user completes the intended task) or ``unsuccessful.`` Based on previous data, the probability of a single interaction being successful is $0.85$ , and user interactions are independent.

Example

A fair six-sided die is rolled twice. Let $X$ be the random variable representing the sum of the two rolls. The probability mass function of $X$ can be calculated by considering all possible outcomes of the two rolls and their associated probabilities.

Example

In a biological research lab, scientists are studying the viability of two types of seeds in a controlled environment. Suppose the probability that a seed from species A germinates successfully is $0.85$ , and the probability that a seed from species B germinates successfully is $0.92$ . Assume that the germination of seeds from the two species is independent.

Definition

Expected Value

Example

Solution

Example

Solution

Remark

Alternatively:

Let
$X_1=$ the number of dots observed on the first die
$X_2=$ the number of dots observed on the second die

$X=$ the sum of the dots observed on the two dice.

Then, $$\mathbb{E}[X] = \mathbb{E}[X_1]+\mathbb{E}[X_2] = 3.5 + 3.5 = 7$$

Example

In the 1990's a nudity war broke out on Brazilian TV. One network skyrocketed to top ratings with prime-time soap operas featuring full-frontal nudity, while another countered with uncut nudity in films. Meanwhile, a third network decided to keep its clothes on and air high-quality literary adaptations instead-earning it the honor of having the worst ratings in TV history.

Let $x$ be the number of soap operas that a Brazilian person watches per week. Based on a sample survey of adults the following probability distribution was prepared:

$$\begin{array}{c|ccccc} \hline x & 0 & 1 & 2 & 3 & 4 & 5 \\ \hline P(X=x) & 0.36 & 0.24 & 0.18 & 0.10 & 0.07 & 0.05 \\ \hline \end{array}$$

Example

The Marquis de Favras was a French aristocrat and staunch supporter of the royal family during the French Revolution. Branded an enemy of the state he was sent to guillotine. Upon reading his death warrant, he quipped ``I see that you have made three spelling mistakes``. Let $X$ represent the number of spelling mistakes in a randomly selected document. The probability distribution of $X$ is given by:

$$\begin{array}{l|ccccccccc}\hline x & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline P(X=x) & 0.05 & 0.10 & 0.20 & 0.25 & 0.15 & 0.10 & 0.08 & 0.05 & 0.02 \\ \hline \end{array}$$

Example

In 1932, the 'Emu War' in Western Australia saw the army deploy mounted machine guns against thousand-strong herds of rampaging thirsty emus. The emus won.

The probability that a random emu will run away from a human is $0.7$

Expected Value of a Sum:
If $X$ and $Y$ are two random variables, the expected value of their sum is: $$\mathbb{E}[X+Y]=\mathbb{E}[X]+\mathbb{E}[Y]$$ This is true regardless of whether $X$ and $Y$ are independent.

For more variables: $$\mathbb{E}[X_1+X_2+\cdots+X_n]=\mathbb{E}[X_1]+\mathbb{E}[X_2]+\cdots+\mathbb{E}[X_n]$$
Linear Transformation:
If $X$ is a random variable and $a$ and $b$ are constants, then: $$\mathbb{E}(aX+b)=a\mathbb{E}[X]+b$$
Expected Value of a Constant:
If $X$ is a random variable and $k$ is a constant, then: $$\mathbb{E}[k]=k$$

Example

A café sells two types of desserts: cookies and cakes. The random variable $X$ represents the revenue from cookie sales in a day (in dollars), while $Y$ represents the revenue from cake sales. Suppose the following information is given:

$\mathbb{E}[X]=50$ : The expected daily revenue from cookies is $\$ 50$.
$\mathbb{E}[X]=80$ : The expected daily revenue from cakes is $\$ 80$.

$X$ and $Y$ are independent.

The café owner wants to analyze their daily revenue, which includes transformations and combinations of these random variables.

Example

A bakery sells two types of products: bread and cakes. The random variable $B$ represents the daily profit (in dollars) from bread, and $C$ represents the daily profit from cakes. The bakery manager has the following information:

$\mathbb{E}[B]=200$ : The expected daily profit from bread is $\$ 200$.
$\mathbb{E}[C]=150$ : The expected daily profit from cakes is $\$ 150$.

Definition

Variance of a Discrete Random Variable

Theorem:
Variance
The variance of a discrete random variable $X$ can also be expressed as:$$\operatorname{Var}(X)=\mathbb{E}[X^2]-(\mathbb{E}[X])^2$$

Definition

Standard Deviation of a Discrete Random Variable

Example

Consider a discrete random variable $X$ with the following probability distribution:$$\begin{array}{c|c|c}x&P(X=x)\\\hline 1&0.2\\2&0.3\\3&0.5\end{array}$$

Example

Consider a discrete random variable $Y$ with the following probability distribution:$$\begin{array}{c|c|c}y&P(Y=y)\\\hline 0&0.1\\1&0.2\\2&0.3\\3&0.4\end{array}$$

Example

A random variable $X$ represents the number of successful tasks completed by a robot in a day. The probability distribution is as follows: $$\begin{array}{c|c|c}x&P(X=x)\\\hline 0&0.1\\1&0.2\\2&0.3\\3&0.2\\4&0.1\\5&0.1\end{array}$$

Theorem:
Variance of a Sum (Independent Variables)
If $X$ and $Y$ are independent random variables, the variance of their sum is the sum of their variances:$$\operatorname{Var}(X+Y)=\operatorname{Var}(X)+\operatorname{Var}(Y)$$ Independence ensures that there is no covariance term.
Theorem:
Variance of a Sum (Dependent Variables)
If $X$ and $Y$ are dependent random variables, the variance of their sum is:$$\operatorname{Var}(X+Y)=\operatorname{Var}(X)+\operatorname{Var}(Y)+2\operatorname{Cov}(X,Y)$$ where $\operatorname{Cov}(X,Y)$ is the covariance between $X$ and $Y$.
Theorem:
Variance of a Constant Times a Random Variable
For a random variable $X$ and a constant $a$, the variance of the product of $a$ and $X$ is:$$\operatorname{Var}(aX)=a^2\operatorname{Var}(X)$$
Theorem:
Variance of a Linear Combination
For random variables $X$ and $Y$, and constants $a$ and $b$, the variance of the linear combination $aX+bY$ is:$$\operatorname{Var}(aX+bY)=a^2\operatorname{Var}(X)+b^2\operatorname{Var}(Y)+2ab\operatorname{Cov}(X,Y)$$ where $\operatorname{Cov}(X,Y)$ is the covariance between $X$ and $Y$.
Theorem:
Variance of a Constant Random Variable
The variance of a constant random variable $X=k$ is zero:$$\operatorname{Var}(k)=0$$

Example

A bakery sells cakes and cookies, and the sales for each are modeled as random variables:

$X$ : The daily revenue from cakes (in dollars), with $\mathbb{E}[X]=50$ and $\operatorname{Var}(X)=25$
$Y$ : The daily revenue from cookies (in dollars), with $\mathbb{E}[Y]=30$ and $\operatorname{Var}(Y)=16$.

Assume the revenues from cakes and cookies are independent.

Example

A café tracks its daily revenue from coffee and muffins, which are modeled as random variables:

$C$ : The daily revenue from coffee sales (in dollars), with $\mathbb{E}[C]=80$ and $\operatorname{Var}(C)=36$
$M$ : The daily revenue from muffin sales (in dollars), with $\mathbb{E}[M]=50$ and $\operatorname{Var}(M)=25$.

The sales of coffee and muffins are dependent, with a covariance of $\operatorname{Cov}(C,M)=12$.

Example

A biologist is studying the relationship between sunlight exposure and plant growth. The growth of a plant ( $G$, in centimeters per week) depends on the hours of sunlight ( $H$, in hours per day) it receives. The relationship is modeled as: $$ G=2 H+5$$

where
$H$ : Hours of sunlight per day, with $\mathbb{E}[H]=6$ and $\operatorname{Var}(H)=1.5$.
$G$ : Plant growth (in $cm /$ week), which is dependent on $H$.

Defintion

Binomial Random Variable

Formula

Mean of A Binomial Random Variable

Formula

Variance and Standard Deviation of A Binomial Random Variable

Long Example 1

On a trip to Finland in 2005, Italian Prime Minister, Silvio Berlusconi, insulted the country by saying that the Finns ate nothing but ``marinated reindeer`` meat and that their cuisine was something to be ``endured`` and not to be enjoyed. Not ones to take the diss lying down, the Finns entered an international pizza competition and won first place! Their winning entry, which featured wild mushrooms and smoked reindeer was called ``Pizza Berlusconi``. Revenge never tasted so good!

A recent poll found that $80\%$ of Finns regularly ordered Pizza Berlusconi when eating out. At a pizza restaurant in Helsinki, six people are about to sit down for lunch.

Example

Ark Encounter, a multi-million-dollar Noah's Ark replica/theme park in Williamstown, Kentucky, is all about surviving floods... theoretically. While the original Ark was designed to withstand 40 days and 40 nights of rain, the Kentucky version didn`t quite live up to the hype. Turns out, after 40-50 inches of rain fell in May '24, the ark itself fell victim to flood damage.

In a twist so ironic that it feels divinely scripted, Ark Encounter's lawyers are now suing their insurers for refusing to cover water damage.

The probability that an insurance company will honour and compensate a policy holder whose property was damaged by flooding is $65 \%$. In the next 30 claims where flooding was the cause of property damage, what is

Example

In 2017, Lee De Paauw from Queensland, Australia tried to impress a girl by jumping into a river after drinking a large quantity of wine. He was immediately mauled by a 3-metre crocodile. And despite managing to fight off the crocdile, he still failed to win a date with the girl of his interest.

Based on historical data, there's a 10% chance of encountering a crocodile when someone enters a specific river. In a bold and questionable move, a group of 15 daredevils has decided to jump into this crocodile-infested river to test their luck.