All Study Guides Financial Mathematics Unit 3
💹 Financial Mathematics Unit 3 – Probability Theory & Random VariablesProbability theory and random variables form the foundation of financial mathematics. These concepts help us understand uncertainty and risk in financial markets, from basic coin flips to complex portfolio management.
Key ideas include probability measures, sample spaces, and events. We explore probability rules, random variables, and distributions. Expected values, variance, and covariance are crucial for analyzing financial data and making informed decisions.
Key Concepts and Definitions
Probability measures the likelihood of an event occurring and ranges from 0 (impossible) to 1 (certain)
Sample space (Ω \Omega Ω ) includes all possible outcomes of an experiment or random process
Event (A) is a subset of the sample space containing one or more outcomes of interest
Mutually exclusive events cannot occur simultaneously in a single trial (rolling a 3 and a 4 on a die)
Independent events do not influence each other's probability (flipping a coin and rolling a die)
P ( A ∩ B ) = P ( A ) ⋅ P ( B ) P(A \cap B) = P(A) \cdot P(B) P ( A ∩ B ) = P ( A ) ⋅ P ( B ) for independent events A and B
Conditional probability P ( A ∣ B ) P(A|B) P ( A ∣ B ) measures the probability of event A given that event B has occurred
Calculated using the formula P ( A ∣ B ) = P ( A ∩ B ) P ( B ) P(A|B) = \frac{P(A \cap B)}{P(B)} P ( A ∣ B ) = P ( B ) P ( A ∩ B )
Probability Basics
Addition rule for mutually exclusive events: P ( A ∪ B ) = P ( A ) + P ( B ) P(A \cup B) = P(A) + P(B) P ( A ∪ B ) = P ( A ) + P ( B )
Addition rule for non-mutually exclusive events: P ( A ∪ B ) = P ( A ) + P ( B ) − P ( A ∩ B ) P(A \cup B) = P(A) + P(B) - P(A \cap B) P ( A ∪ B ) = P ( A ) + P ( B ) − P ( A ∩ B )
Multiplication rule for independent events: P ( A ∩ B ) = P ( A ) ⋅ P ( B ) P(A \cap B) = P(A) \cdot P(B) P ( A ∩ B ) = P ( A ) ⋅ P ( B )
Multiplication rule for dependent events: P ( A ∩ B ) = P ( A ) ⋅ P ( B ∣ A ) P(A \cap B) = P(A) \cdot P(B|A) P ( A ∩ B ) = P ( A ) ⋅ P ( B ∣ A )
Complementary events have probabilities that sum to 1 (rolling an even number and rolling an odd number on a die)
P ( A ) + P ( A c ) = 1 P(A) + P(A^c) = 1 P ( A ) + P ( A c ) = 1 , where A c A^c A c is the complement of event A
Bayes' theorem relates conditional probabilities: P ( A ∣ B ) = P ( B ∣ A ) ⋅ P ( A ) P ( B ) P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)} P ( A ∣ B ) = P ( B ) P ( B ∣ A ) ⋅ P ( A )
Random Variables
Random variable (X) assigns a numerical value to each outcome in a sample space
Discrete random variables have countable outcomes (number of heads in 5 coin flips)
Continuous random variables have uncountable outcomes within an interval (time until a stock reaches a certain price)
Probability mass function (PMF) gives the probability of each value for a discrete random variable
p ( x ) = P ( X = x ) p(x) = P(X = x) p ( x ) = P ( X = x ) , where x is a possible value of X
Probability density function (PDF) describes the probability distribution for a continuous random variable
f ( x ) f(x) f ( x ) is the PDF, and P ( a ≤ X ≤ b ) = ∫ a b f ( x ) d x P(a \leq X \leq b) = \int_a^b f(x) dx P ( a ≤ X ≤ b ) = ∫ a b f ( x ) d x
Probability Distributions
Bernoulli distribution models a single trial with two possible outcomes (success or failure)
P ( X = 1 ) = p P(X = 1) = p P ( X = 1 ) = p and P ( X = 0 ) = 1 − p P(X = 0) = 1 - p P ( X = 0 ) = 1 − p , where p is the probability of success
Binomial distribution models the number of successes in a fixed number of independent Bernoulli trials
P ( X = k ) = ( n k ) p k ( 1 − p ) n − k P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} P ( X = k ) = ( k n ) p k ( 1 − p ) n − k , where n is the number of trials and k is the number of successes
Poisson distribution models the number of events occurring in a fixed interval of time or space
P ( X = k ) = e − λ λ k k ! P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!} P ( X = k ) = k ! e − λ λ k , where λ \lambda λ is the average rate of events
Normal (Gaussian) distribution is a continuous distribution with a bell-shaped curve
Characterized by its mean (μ \mu μ ) and standard deviation (σ \sigma σ )
Standard normal distribution has μ = 0 \mu = 0 μ = 0 and σ = 1 \sigma = 1 σ = 1
Expectation and Variance
Expected value (mean) of a random variable is the average value over many trials
For discrete X: E ( X ) = ∑ x x ⋅ p ( x ) E(X) = \sum_{x} x \cdot p(x) E ( X ) = ∑ x x ⋅ p ( x )
For continuous X: E ( X ) = ∫ − ∞ ∞ x ⋅ f ( x ) d x E(X) = \int_{-\infty}^{\infty} x \cdot f(x) dx E ( X ) = ∫ − ∞ ∞ x ⋅ f ( x ) d x
Variance measures the average squared deviation from the mean
V a r ( X ) = E [ ( X − μ ) 2 ] = E ( X 2 ) − [ E ( X ) ] 2 Var(X) = E[(X - \mu)^2] = E(X^2) - [E(X)]^2 Va r ( X ) = E [( X − μ ) 2 ] = E ( X 2 ) − [ E ( X ) ] 2
Standard deviation is the square root of the variance: σ = V a r ( X ) \sigma = \sqrt{Var(X)} σ = Va r ( X )
Linearity of expectation: E ( a X + b Y ) = a E ( X ) + b E ( Y ) E(aX + bY) = aE(X) + bE(Y) E ( a X + bY ) = a E ( X ) + b E ( Y ) for constants a and b
Covariance measures the linear relationship between two random variables X and Y
C o v ( X , Y ) = E [ ( X − E ( X ) ) ( Y − E ( Y ) ) ] Cov(X, Y) = E[(X - E(X))(Y - E(Y))] C o v ( X , Y ) = E [( X − E ( X )) ( Y − E ( Y ))]
Applications in Finance
Portfolio return is a weighted average of individual asset returns
R p = ∑ i = 1 n w i R i R_p = \sum_{i=1}^n w_i R_i R p = ∑ i = 1 n w i R i , where w i w_i w i is the weight of asset i and R i R_i R i is its return
Portfolio variance depends on asset variances and covariances
σ p 2 = ∑ i = 1 n w i 2 σ i 2 + 2 ∑ i = 1 n ∑ j = i + 1 n w i w j σ i σ j ρ i j \sigma_p^2 = \sum_{i=1}^n w_i^2 \sigma_i^2 + 2 \sum_{i=1}^n \sum_{j=i+1}^n w_i w_j \sigma_i \sigma_j \rho_{ij} σ p 2 = ∑ i = 1 n w i 2 σ i 2 + 2 ∑ i = 1 n ∑ j = i + 1 n w i w j σ i σ j ρ ij
ρ i j \rho_{ij} ρ ij is the correlation coefficient between assets i and j
Value at Risk (VaR) estimates the potential loss for an investment over a given time horizon and confidence level
For a normal distribution: V a R = μ + z α σ VaR = \mu + z_{\alpha} \sigma Va R = μ + z α σ , where z α z_{\alpha} z α is the z-score for the desired confidence level
Option pricing models (Black-Scholes) use probability distributions to estimate the fair price of options contracts
Common Probability Problems
Gambler's Ruin problem analyzes the probability of a gambler going bankrupt
Depends on initial capital, win probability, and bet size
Birthday Problem calculates the probability of two people sharing a birthday in a group
Surprisingly high for relatively small groups (50% for 23 people)
Monty Hall problem demonstrates the importance of conditional probability
Switching doors after the host reveals a goat increases the win probability to 2/3
Coupon Collector's problem determines the expected number of trials to collect all unique items
E ( X ) = n ( 1 n + 1 n − 1 + . . . + 1 2 + 1 ) E(X) = n(\frac{1}{n} + \frac{1}{n-1} + ... + \frac{1}{2} + 1) E ( X ) = n ( n 1 + n − 1 1 + ... + 2 1 + 1 ) , where n is the number of unique items
Combinatorics for counting possibilities
Permutations: n P r = n ! ( n − r ) ! nPr = \frac{n!}{(n-r)!} n P r = ( n − r )! n !
Combinations: n C r = ( n r ) = n ! r ! ( n − r ) ! nCr = \binom{n}{r} = \frac{n!}{r!(n-r)!} n C r = ( r n ) = r ! ( n − r )! n !
Moment generating functions (MGFs) uniquely characterize probability distributions
M X ( t ) = E ( e t X ) = ∑ x e t x ⋅ p ( x ) M_X(t) = E(e^{tX}) = \sum_{x} e^{tx} \cdot p(x) M X ( t ) = E ( e tX ) = ∑ x e t x ⋅ p ( x ) for discrete X
M X ( t ) = E ( e t X ) = ∫ − ∞ ∞ e t x ⋅ f ( x ) d x M_X(t) = E(e^{tX}) = \int_{-\infty}^{\infty} e^{tx} \cdot f(x) dx M X ( t ) = E ( e tX ) = ∫ − ∞ ∞ e t x ⋅ f ( x ) d x for continuous X
Law of Large Numbers states that the sample mean approaches the population mean as the sample size increases
lim n → ∞ P ( ∣ X ˉ n − μ ∣ < ϵ ) = 1 \lim_{n \to \infty} P(|\bar{X}_n - \mu| < \epsilon) = 1 lim n → ∞ P ( ∣ X ˉ n − μ ∣ < ϵ ) = 1 for any ϵ > 0 \epsilon > 0 ϵ > 0
Central Limit Theorem asserts that the sum of many independent random variables is approximately normally distributed
Useful for approximating probabilities for large samples
Markov Chains model systems transitioning between states with fixed probabilities
Transition matrix P contains the probabilities of moving from one state to another