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LINEAR ALGEBRA /// STATISTICS /// 30 MIN READ

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1. Joint Probability Mass Functions

When analyzing two discrete random variables simultaneously, we map out their intersections using a Joint PMF table. A valid joint PMF must sum to $1$ across all cells.

The Setup: A fair coin is tossed three times. Let $X$ denote the number of heads obtained and let $Y$ be defined as the toss on which the first head appears (or $0$ if no heads). Construct the joint PMF of $X$ and $Y$.

By mapping the 8 possible outcomes ($2^3$), we evaluate $X$ (total heads) and $Y$ (first appearance) for each:

TTT $\rightarrow X=0, Y=0$
HTT $\rightarrow X=1, Y=1$
THT $\rightarrow X=1, Y=2$
HHT $\rightarrow X=2, Y=1$
... (and so on)

Y \ X	0	1	2	3
0	1/8	0	0	0
1	0	1/8	2/8	1/8
2	0	1/8	1/8	0
3	0	1/8	0	0

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2. Conditional Probability

Conditional probabilities allow us to slice the joint PMF based on known constraints. The fundamental equation is:

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

Exam Question: The joint PMF of two discrete random variables $X$ and $Y$ is given by: $$ f_{XY}(x,y) = \begin{cases} \frac{1}{36}(3x + y), & x, y \in \{0, 1, 2\} \\ 0, & \text{otherwise} \end{cases} $$ What is the value of $P(0 < X < 2 \mid Y > 1)$?

Step 1: Simplify the conditions.

For integers, $0 < X < 2$ means exactly $X = 1$. The condition $Y > 1$ means exactly $Y = 2$. So we need $P(X = 1 \mid Y = 2)$.

Step 2: Find the joint probability (numerator).

$$ f_{XY}(1,2) = \frac{1}{36}(3(1) + 2) = \frac{5}{36} $$

Step 3: Find the marginal probability (denominator).

Sum across all $X$ values where $Y=2$:

$$ P(Y=2) = f(0,2) + f(1,2) + f(2,2) = \frac{2}{36} + \frac{5}{36} + \frac{8}{36} = \frac{15}{36} $$

✅ FINAL SOLUTION

$$ P(X=1 \mid Y=2) = \frac{5/36}{15/36} = \frac{5}{15} = 0.33 $$

GEOMETRIC ALGEBRA

3. Linear Systems Geometry

A system of linear equations isn't just numbers—it represents the intersection of geometric planes in space.

Unique Solution: Planes intersect at exactly one point (e.g., $x+y=3, x-y=-3$).
Infinite Solutions: Planes intersect along a continuous line (e.g., $z=0, x+y+z=1$).
No Solution: Parallel planes that never meet (e.g., $x+y+z=1, x+y+z=7$).

Fig 1. Planes in $\mathbb{R}^3$

HACKING DETERMINANTS

Calculating determinants by brute force is for amateurs. Use properties to break down complex expressions instantly.

$\det(AB) = \det(A)\det(B)$

$\det(A^k) = (\det(A))^k$

$\det(A^{-1}) = \frac{1}{\det(A)}$

$\det(cA) = c^n \det(A)$ (n = order)

Example: Let $A$ be a $3 \times 3$ matrix with $\det(A) = 5$. Let $B$ be a matrix obtained by adding 8 times Row 1 of $A$ to Row 3, and 9 times Row 2 to Row 1. Find $\det(8A^2B^{-1})$.

Step 1: Understand Row Operations

Adding a multiple of one row to another does not change the determinant. Therefore, $\det(B) = \det(A) = 5$.

Step 2: Expand the target expression

$$ \begin{aligned} \det(8A^2B^{-1}) &= 8^3 \times \det(A^2) \times \det(B^{-1}) \\ &= 512 \times (\det(A))^2 \times \frac{1}{\det(B)} \end{aligned} $$

🚀 CRUNCHING THE NUMBERS

$$ 512 \times (5)^2 \times \frac{1}{5} = 512 \times 25 \times 0.2 = 2560 $$

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