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 Mean
Key points
- Understand the fundamental concepts of probability.
- Identify and apply common probability distributions.
- Calculate and interpret descriptive statistics.
- 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
Mean (Expected Value)
- Meaning
The average value of a random variable over many trials.
- Term
Variance
- Meaning
A measure of the spread or dispersion of a set of data or a probability distribution.
- Term
Conditional Probability
- Meaning
The probability of an event occurring given that another event has already occurred.
- Term
Central Limit Theorem
- Meaning
A statistical theorem stating that the distribution of sample means approximates a normal distribution as the sample size becomes large.
Learning objectives
Understand the fundamental concepts of probability.
Identify and apply common probability distributions.
Calculate and interpret descriptive statistics.
Apply Bayes' Theorem for conditional probability.
Understand the principles of statistical inference and hypothesis testing.
Formulae
- Name
Binomial Distribution Mean
- Note
n = number of trials, p = probability of success
- Expression
E[X] = np
- Name
Binomial Distribution Variance
- Note
n = number of trials, p = probability of success
- Expression
Var(X) = np(1-p)
- Name
Poisson Distribution Mean
- Note
λ = average rate of occurrence
- Expression
E[X] = λ
- Name
Poisson Distribution Variance
- Note
λ = average rate of occurrence
- Expression
Var(X) = λ
- Name
Normal Distribution PDF
- Note
μ = mean, σ = standard deviation
- Expression
f(x) = (1/(σ√(2π))) * e^(-(x-μ)²/(2σ²))
- Name
Bayes' Theorem
- Note
Used to update probability based on new evidence
- Expression
P(A|B) = [P(B|A)P(A)] / P(B)
- Name
Expected Value (Discrete)
- Note
Sum over all possible values xᵢ
- Expression
E[X] = Σ xᵢP(xᵢ)
- Name
Variance (Discrete)
- Note
Measure of spread
- Expression
Var(X) = E[(X - E[X])²] = Σ (xᵢ - E[X])²P(xᵢ)
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.
Assuming normality without justification.
Errors in calculating variance and standard deviation.
Keywords
Probability
Statistics
Random Variable
Distribution
Mean
Variance
Bayes' Theorem
Central Limit Theorem
Hypothesis Testing
Regression
Practice preview
A fair coin is tossed three times. What is the probability of getting exactly two heads?…
easy
If A and B are two mutually exclusive events, and P(A) = 0.4, P(B) = 0.3, what is P(A U B)?…
easy
A bag contains 5 red and 3 blue balls. If two balls are drawn without replacement, what is the probability that the first ball is red and the second ball is blue?…
medium