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University of Toronto St. George
Statistical Sciences
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STA257H1a.doc R ANDOM V ARIABLES AND E VENTS Definition: Random Variable A random variable is a variable that is random. They are denoted X, Y, Z. Note A random variable which is constant will be called a constant random variable. Example Roll a fair die and let X denote the number of dots. X is a random variable. Its possible values are 1, 2, 3, 4, 5, 6. Observing X leads to data or observed value. We will denote a typical observation by x. Definition: Event In general, an event a statement involving random variables. They are denoted A, B, C, D. Notation {X is even} means the event that X is even. Note Events either occur or they dont. Notes 1) Events may be thought of as collectionssets of outcomes. 2) The sure event (denoted S or ) is the set of all possible outcomes. 3) S includes all the outcomes so that any event A is made up of outcomes drawn from S. A is a subset of S and we write A S . 4) The impossible event or empty set (denoted ) is the event which consists of no possible outcomes and hence never occurs. Notice: A S . 2 X 5) We can talk about any function of X, sa(X ). Examples: X , sin X ,e = exp(X), etc. Definition: The Indicator Random Variable or Bernoulli Random Variable If A is an event, then we define the indicator random variable or Bernoulli random variable forof A to be if Aoccurs IA = . 0otherwise Example Let X be a random variable. Measure it an infinite number of times 1o 2et 3 , x , x ,. The average will be the expected value or expectation of X, de(Xt). E If X denotes marks, then A{pas} ={X 0}. The indicator random variable might look like: 0,0,1,0,. Page 1 of 25 STA257H1a.doc Definition The proportion or the relative frequency of 1s will be the probability of (A).enoted P Notice P(A )= E(IA). M ORE O N E VENTS AND R ANDOM VARIABLES Events consist of outcomes. Every event is a subset of S. Sometimes the same event has different descriptions, so we might want to show something like: A B A = B . B A How to show A B? Take an arbitrary element of A. Show that it is an element of B, ix Aleand show xB . From our point of view, B means A B . Two events are equal (i.e. the samA) ifB and B A . Random Variables How to show two random variables and Y are equal? They are equal if they are equal all the time. Example Let A and A be disjoint events (i.e. no outcomes in common, no overlap, etc.). Look aA A event 1 2 1 2 (the event that at least one of thecurs). The indicator random variable of this new eveIt is or I(A A ). Clearly, A1 2 1 2 I = I + I . A1 2 A1 A2 Notes 1) A = A A is the event that at least one of theurs. i 1 2 i1 2) If the As are disjoint, then a common notationA = A + A + . i 1 2 i=1 3) A i A 1 A 2 = A1A2 is the event that all of the As occur. i1 C OMPLEMENTS Definition c The complement of an event A is the event consisting of all outcomes not in A. It is denoted A or A . Page 2 of 25
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