note 3 - MA2216/ST2131 Probability Notes 3 § 1. Random...

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Unformatted text preview: MA2216/ST2131 Probability Notes 3 § 1. Random Variables. It is frequently the case that, when an experiment is performed, we are mainly interested in some function of the outcome as opposed to the actual outcome itself. For instance, in testing 100 electronic components, we are often con- cerned with the number of defectives that occur. Also, in coin-flipping, we may be interested in the total number of heads that occur and not care at all about the actual head-tail sequence that results. These values are, of course, random quantities determined by the out- come of the experiment. 1. Definition: A random variable ( r.v. ) is a real-valued function de- fined on the sample space. Usually, r.v.’s are denoted by X , Y , Z , etc. We say X is a random variable if X : S → IR is a mapping on S , taking real values. A random variable is said to be discrete if the range of X is either finite or countably infinite. We will focus on discrete r.v.’s in § 2. 1 2. Examples: (a) Toss a pair of fair dice. Let X be the sum of the upturned faces. Obviously, X is a discrete r.v., taking values in { 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 } . (Refer to Ex. 5 of Tutorial 1.) (b) A die is thrown until a 5 occurs. Let X be the number of trials required. Clearly, X can take from 1 , 2 , 3 ,... . (c) You are waiting for SBS 97 at a bus stop. Let X be the waiting time in minutes at the bus stop. Since the range of X is (0 , ∞ ), X is not a discrete r.v. (d) There are 20 questions in a multiple choice paper. Each question has 5 alternatives. A student answers all 20 questions by randomly and independently choosing one alternative out of 5 in each question. Let X denote the number of correct answers. 3. Distribution Function & its Properties The cumulative distribution function ( c.d.f. ) (simply distribution function ( d.f. )) F X of X is defined by F X ( x ) = IP { X ≤ x } ,-∞ < x < ∞ . Before proceeding, let us take a look at the notation { X ≤ x } . It is an event which contains all sample points s such that X ( s ) ≤ x . That is, { X ≤ x } = { s ∈ S : X ( s ) ∈ (-∞ , x ] } = X- 1 ((-∞ , x ]) , the inverse image of the semi-infinite (closed) interval (-∞ , x ]. 2 Analytically , every distribution function F X ( · ) satisfies the following properties: 4. Properties: Let X be a random variable with F X its distribution func- tion. Then (i) 0 ≤ F X ( x ) ≤ 1 for all x . (ii) F X is non-decreasing, i.e., F X ( a ) ≤ F X ( b ) if a < b. Observe that, for a < b , (-∞ , a ] ⊂ (-∞ , b ] , and hence F X ( a ) = IP { X ∈ (-∞ , a ] } ≤ IP { X ∈ (-∞ , b ] } = F X ( b ) . (iii) lim x →-∞ F X ( x ) = 0, lim x →∞ F X ( x ) = 1 ....
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note 3 - MA2216/ST2131 Probability Notes 3 § 1. Random...

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