2025-11-20
Definition
Statistics is the science of collecting, organizing, analyzing, interpreting, and presenting data. Used in research, business, science, and everyday decision-making
Two kinds of statistics:
Definition
Probability is the measure of the likelihood that an event will occur.
Formula for Probability
\[P(\text{Event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}=\frac{x}{n}\]
Example: Rolling a die
What is the probability of rolling a 4 in a die?
Definition
A sample space (S) is the collection or set of all possible outcomes of a random experiment.
Examples:
Definition
An event is a subset of possible outcomes from the sample space.
Important
When you calculate the probability of the event \(P(E)\), it is ALWAYS a decimal number, a fractional number, ranges from 0 to 1.
For example:
When tossing a coin, what’s the probability of getting heads? There should be two outcomes: getting a head & getting a tail . The sample space would be: \(S=\{H,\ T\}\) And the event A is getting head Thus, \(x=1;\ n=2\), and the probability would be:
\[P(A)=\frac{1}{2}=0.5\]
Definition
An event is any subset of possible outcomes in the sample space \(\Omega\).
Compound event: contains two or more outcomes (e.g., “rolling an even number”)
Practice Problem:
A bag contains 5 red balls, 3 blue balls, and 2 green balls.
Two events \(A\) and \(B\) are mutually exclusive if they have no outcomes in common:
\[A \cap B = \emptyset\]
Consequence for probabilities:
\[P(A \cup B) = P(A) + P(B)\] (no overlap to double-count)
Classic example – Die roll
\(\rightarrow\) \(A \cap B = \emptyset\) \(\rightarrow\) mutually exclusive
Another example: Drawing one card
\(\rightarrow\) also mutually exclusive
Two events \(A\) and \(B\) are non-mutually exclusive if they share at least one outcome:
\[A \cap B \neq \emptyset\]
Addition rule becomes:
\[P(A \cup B) = P(A) + P(B) - P(A \cap B)\] (we subtract the overlap so it’s not counted twice)
Die roll example
\(\rightarrow\) \(A \cap B = \{6\} \neq \emptyset\) \(\rightarrow\) not mutually exclusive
\(P(A \cup B) = P(A) + P(B) - P(A \cap B) = \frac{3}{6} + \frac{2}{6} - \frac{1}{6} = \frac{4}{6} = \frac{2}{3}\)
Definition
A random variable is a function that assigns a numerical value to each outcome in the sample space of a random experiment.
Notation: We use capital letters (X, Y, Z) for random variables
How it works:
Intuition: Tossing two coins
Definition
A random variable is a function that assigns numerical values to outcomes of a random experiment.
Example
Rolling a die: \(X = \text{number shown}\)
Takes on countable values
Key Property: Can list all discrete possible values
Examples:
Takes on uncountably infinite values
Key Property: Any value in an interval is possible
Examples:
A probability distribution assigns probabilities to each possible value.
Requirements:
Example: Rolling a die
| \(X\) | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| \(P(X)\) | 1/6 | 1/6 | 1/6 | 1/6 | 1/6 | 1/6 |
The expected value E(X) is the average value we expect from a random variable.
\[E(X) = \sum_{i}^{\infty} x_i \cdot P(X = x_i)\]
\[E(X) = 1 \cdot \frac{1}{6} + 2 \cdot \frac{1}{6} + 3 \cdot \frac{1}{6} + 4 \cdot \frac{1}{6} + 5 \cdot \frac{1}{6} + 6 \cdot \frac{1}{6}\]
\[E(X) = \frac{21}{6} = 3.5\]
Measures how spread out the values are from the mean.
\[\text{Var}(X) = E[(X - \mu)^2]\] \[= \sum_{i} (x_i - \mu)^2 \cdot P(X = x_i)\]
where \(\mu = E(X)\)
Measures how spread out the values are from the mean.
\[\text{Var}(X) = E[(X - \mu)^2]\] \[= \sum_{i} (x_i - \mu)^2 \cdot P(X = x_i)\]
where \(\mu = E(X)\)
The square root of variance (same units as X).
\[\sigma = \sqrt{\text{Var}(X)}\]
Interpretation:
Many real-life situations follow repeatable patterns. When we study those patterns carefully, we discover a handful of probability distributions that keep showing up again and again.
The normal distribution (the bell curve) is the most famous distribution out there. Yes, it is only a member of a much larger family.
Here are common discrete probability distribution.
Scenario: A single trial with only two possible outcomes (success or failure)
\[P(X = x) = \begin{cases} p & \text{if } x = 1 \text{ (success)} \\ 1-p & \text{if } x = 0 \text{ (failure)} \end{cases}\]
\[P(X = x)=p^x(1-p)^{1-x}\]
Properties: - \(E(X) = p\) - \(\text{Var}(X) = p(1-p)\)
Example: Flipping a coin once, \(X = 1\) if heads, \(X = 0\) if tails - \(p = 0.5\) - \(E(X) = 0.5\) - \(\text{Var}(X) = 0.5(0.5) = 0.25\)
Scenario: n independent trials, each with probability p of success
\[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\]
Example: Flipping a coin 10 times, X = number of heads
Problem: A bag contains 3 red balls and 2 blue balls. You draw 2 balls without replacement. Let X = number of red balls drawn.
Find:
Probability Distribution:
Expected Value:
E(X) = 0(1/10) + 1(3/5) + 2(3/10) = 0 + 3/5 + 6/10 = 12/10 = 1.2
Variance:
Var(X) = (0-1.2)²(1/10) + (1-1.2)²(3/5) + (2-1.2)²(3/10) = 0.36