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Nigeria's free digital learning hub. Videos, lecture notes, practice questions and leaderboards aligned with CCMAS, NBTE, NCCE, WAEC, NECO, IJMB and JAMB curricula.
Statistics · 400 Level
This is the complete lecture note for STA 412 Quadratic forms and the fisher cochran theorem, prepared for Statistics students at 400 Level. You can read it online below, open it in the full viewer, or download the PDF for offline study. It pairs with the course's video lectures, so you can read and watch together.
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS 1 Probability Spaces, Measures, and Distributions Series Outline • 1.1 Probability Spaces • 1.1.1 Definition of probability space • 1.1.2 Sample space, events, and sigma-algebra • 1.2 Probability Measures • 1.2.1 Axioms of probability • 1.2.2 Properties of probability measures • 1.3 Distribution of Random Variables as Measurable Functions • 1.3.1 Random variables and measurability • 1.3.2 Probability distribution induced by random variables • 1.4 Product Spaces and Product Probabilities • 1.4.1 Product of measurable spaces • 1.4.2 Product probability measures • 1.5 Applications and Illustrations • 1.5.1 Worked examples of probability spaces • 1.5.2 Case study: product probabilities in practice Introduction Probability theory provides the foundation for much of modern statistics and data science. At its core lies the concept of a probability space, which formalises the idea of randomness and uncertainty. By defining sample spaces, events, and probability measures, we can systematically study random phenomena. This Series is crucial because it sets the stage for all subsequent topics in STA 411, giving you the tools to describe and analyse random variables, their distributions, and the structures that arise when combining multiple spaces. The aim of this Series is to introduce you to the formal framework of probability spaces, measures, and distributions, and to equip you with the ability to apply these concepts to random variables and product spaces.
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS We shall commence this Series by examining probability spaces, including sample spaces, events, and sigma-algebras. Then, we will move on to probability measures, where we will study the axioms and properties that govern them. Subsequently, we will explore random variables as measurable functions and the distributions they induce. Following that, we will consider product spaces and product probabilities, which extend the framework to multiple random variables. Finally, we will wrap up with applications and illustrations that demonstrate how these concepts are used in practice. Series Learning Outcomes Upon completing this Series, you should be able to: • SLO 1.1 define a probability space and describe its components including sample space, events, and sigma-algebra, • SLO 1.2 explain the axioms and properties of probability measures, • SLO 1.3 demonstrate how random variables can be treated as measurable functions and how they induce probability distributions, • SLO 1.4 analyse product spaces and construct product probability measures, and • SLO 1.5 apply the concepts of probability spaces, measures, and distributions to worked examples and case studies. 1.1 Probability Spaces Probability theory begins with the concept of a probability space, which provides the formal framework for studying random phenomena. A probability space consists of three elements: a sample space, a collection of events, and a probability measure. Together, these elements allow us to assign numerical values to the likelihood of different outcomes. 1.1.1 Definition of probability space A probability space is defined as a triple P, where: • is the sample space, the set of all possible outcomes of a random experiment. • Fis a sigma-algebra, a collection of subsets of called events, which satisfies certain properties such as closure under union and complement.
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS • Pis the probability measure, a function that assigns a probability between 0 and 1 to each event in F. Fill in the blank: A probability space is defined as a triple ____. Components of a probability space. 1.1.2 Sample space,events, and sigma-algebra The sample space is the set of all possible outcomes of an experiment. For example, when tossing a coin, the sample space is ={H,T}. An event is a subset of the sample space. For instance, the event “getting a head” is H. A sigma-algebra is a collection of events that includes the sample space itself, is closed under complementation, and closed under countable unions. This ensures that probability can be consistently assigned to events. Fill in the blank: A sigma-algebra must be closed under complementation and __________ unions. Properties of a sigma-algebra • Contains the sample space • Closed under complementation • Closed under countable unions • Closed under countable intersections
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Pilot Questions 1.1 1. A probability space is defined as: A. PB. FC. FD. None of the above 2. The set of all possible outcomes of a random experiment is called the: A. Event B. Sample space C. Sigma-algebra D. Probability measure 3. Which of the following is an event when tossing a coin? A. HB. TC. TD. All of the above 4. A sigma-algebra must be closed under: A. Addition and subtraction B. Complementation and countable unions C. Multiplication and division D. None of the above 5. In the probability space P, the function that assigns numerical values to events is the: A. Sample space B. Sigma-algebra C. Probability measure D. Event 1.2 Probability Measures Probability measures are the mathematical rules that allow us to assign numerical values to events within a probability space. They ensure that probabilities are consistent, logical, and mathematically valid. Without probability measures, the framework of probability spaces would remain abstract and incomplete. 1.2.1 Axioms of probability A probability measure is a function P:F→[0,1]that assigns a probability to each event in the sigma-algebra F. It must satisfy three fundamental axioms: 1. Non-negativity: For any event A∈F, P(A)≥0. 2. Normalisation: The probability of the entire sample space is 1, i.e. P()=1. 3. Countable additivity: If A1,A2,A3,…are disjoint events, then P⋃i=1Ai=i=1P(Ai). Fill in the blank: The probability of the entire sample space must equal __________. Axioms of probability • Non-negativity: P(A)≥0 • Normalisation: P()=1
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS • Countable additivity: P(∪Ai)=∑P(Ai) 1.2.2 Properties of probability measures From these axioms, several useful properties can be derived: • Probability of the empty set: P(∅)=0. • Boundedness: For any event A, 0≤P(A)≤1. • Complement rule: For any event A, P(Ac)=1-P(A). • Monotonicity: If A⊆B, then P(A)≤P(B). • Inclusion-exclusion principle: For two events Aand B, P(A∪B)=P(A)+P(B)-P(A∩B). Fill in the blank: If A⊆B, then P(A) \leq __________. Properties of probability measures.
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS Pilot Questions 1.2 1. Which of the following is one of the axioms of probability? A. P()=1B. P()=0C. P(∅)=1D. None of the above 2. The probability of the empty set is: A. 1 B. 0 C. Undefined D. Depends on the measure 3. If A⊆B, then: A. P(A)≥P(B)B. P(A)≤P(B)C. P(A)=P(B)D. None of the above 4. The complement rule states that: A. P(Ac)=1-P(A)B. P(Ac)=P(A)C. P(Ac)=0D. None of the above 5. For two events Aand B, the inclusion-exclusion principle is: A. P(A∪B)=P(A)+P(B)B. P(A∪B)=P(A)+P(B)-P(A∩B)C. P(A∪B)=P(A)-P(B)D. None of the above 1.3 Distribution Of Random Variables As Measurable Functions Random variables are the bridge between abstract probability spaces and practical applications. They allow us to assign numerical values to outcomes of experiments and then study the resulting distributions. In this section, we explore random variables as measurable functions and how they induce probability distributions. 1.3.1 Random variables and measurability A random variable is a function X:Rthat assigns a real number to each outcome in the sample space . For a function to qualify as a random variable, it must be measurable. This means that for any real number x, the set ω∈:X(ω)≤x must belong to the sigma-algebra F. Measurability ensures that probabilities can be consistently assigned to events defined by the random variable. Example: If Xrepresents the outcome of rolling a die, then the event 3corresponds to the subset 3of the sample space. Fill in the blank: A random variable must be __________ with respect to the sigma-algebra of the probability space. 1.3.2 Probability distribution induced by random variables
LA-NET | ONLINE LEARNING VIDEO PLATFORM FOR POST-SECONDARY & HIGHER INSTITUTION STUDENTS The probability distribution of a random variable describes how probabilities are assigned to the values that the random variable can take. Formally, if Xis a random variable, its distribution is defined by: PX(B)=P({ω∈:X(ω)∈B}),B⊆R. This means that the distribution of Xis the push-forward of the probability measure Pthrough the function X. Example: If Xrepresents the outcome of rolling a fair die, then the distribution of Xassigns probability 1/6to each of the numbers 1 through 6. Fill in the blank: The distribution of a random variable is the __________ of the probability measure through the function X. Pilot Questions 1.3 1. A random variable is defined as: A. A measurable function from the sample space to real numbers B. A subset of the sample space C. A probability measure D. None of the above 2. For a function to qualify as a random variable, it must be: A. Continuous B. Measurable C. Differentiable D. None of the above 3. The distribution of a random variable is defined by: A. The sigma-algebra B. The push-forward of the probability measure C. The sample space D. None of the above 4. If Xrepresents the outcome
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