Probability and Statistics
What is Probability and Statistics?
A variable whose value is a numerical outcome of a random phenomenon.
Key formula / rule: Binomial Distribution Probability
Key points
- Understand the fundamental concepts of probability.
- Identify and apply appropriate probability distributions.
- Calculate and interpret expected values and variances.
- Apply Bayes' Theorem for conditional probability.
Common exam trap
Confusing independent and dependent events.
Definitions
- Term
Random Variable
- Meaning
A variable whose value is a numerical outcome of a random phenomenon.
- Term
Probability Distribution
- Meaning
A function that describes the likelihood of obtaining the possible values that a random variable can assume.
- Term
Expected Value
- Meaning
The weighted average of all possible values of a random variable, weighted by their probabilities. It represents the long-run average value.
- Term
Variance
- Meaning
A measure of the spread or dispersion of a random variable's values around its expected value.
- Term
Conditional Probability
- Meaning
The probability of an event occurring given that another event has already occurred. Denoted as P(A|B).
- Term
Central Limit Theorem (CLT)
- Meaning
A theorem stating that the distribution of sample means will approximate a normal distribution as the sample size becomes large, regardless of the population's distribution.
Learning objectives
Understand the fundamental concepts of probability.
Identify and apply appropriate probability distributions.
Calculate and interpret expected values and variances.
Apply Bayes' Theorem for conditional probability.
Understand statistical inference techniques like estimation and hypothesis testing.
Analyze relationships between variables using correlation and regression.
Formulae
- Name
Binomial Distribution Probability
- Note
n = number of trials, k = number of successes, p = probability of success
- Expression
P(X=k) = C(n, k) * pk * (1-p)^(n-k)
- Name
Binomial Distribution Expected Value
- Note
- Expression
E[X] = np
- Name
Binomial Distribution Variance
- Note
- Expression
Var(X) = np(1-p)
- Name
Poisson Distribution Probability
- Note
λ = average rate of occurrence, k = number of occurrences
- Expression
P(X=k) = (e^(-λ) * λk) / k!
- Name
Poisson Distribution Expected Value
- Note
- Expression
E[X] = λ
- Name
Poisson Distribution Variance
- Note
- Expression
Var(X) = λ
- Name
Normal Distribution Probability Density Function (PDF)
- Note
μ = mean, σ = standard deviation
- Expression
f(x) = (1 / (σ * √(2π))) * e^(-(x-μ)² / (2σ²))
- Name
Normal Distribution Expected Value
- Note
- Expression
E[X] = μ
- Name
Normal Distribution Variance
- Note
- Expression
Var(X) = σ²
- Name
Bayes' Theorem
- Note
Where P(B) = P(B|A)P(A) + P(B|A')P(A')
- Expression
P(A|B) = [P(B|A) * P(A)] / P(B)
- Name
Covariance
- Note
- Expression
Cov(X, Y) = E[(X - E[X])(Y - E[Y])] = E[XY] - E[X]E[Y]
- Name
Correlation Coefficient
- Note
σX and σY are standard deviations of X and Y
- Expression
ρ(X, Y) = Cov(X, Y) / (σX * σY)
Prerequisites
Basic Algebra
Set Theory
Calculus (Differentiation and Integration)
Common mistakes
Confusing independent and dependent events.
Incorrectly applying formulas for discrete vs. continuous variables.
Misinterpreting conditional probability.
Errors in calculating variance or standard deviation.
Assuming normality without justification (violating Central Limit Theorem conditions).
Keywords
Probability
Statistics
Random Variable
Distribution
Expected Value
Variance
Conditional Probability
Bayes' Theorem
Central Limit Theorem
Hypothesis Testing
Correlation
Regression
Practice preview
If P(A) = 0.4, P(B) = 0.3, and A and B are mutually exclusive events, what is P(A union B)?…
easy
Which of the following statements about the standard normal distribution is INCORRECT?…
medium
A factory produces items using three machines, A, B, and C. Machine A produces 50% of the items, B produces 30%, and C produces 20%. The defect rates for machines A, B, and C are 2%, 3%, and 3% respectively. If a randoml…
medium