×
IT'S TIME TO WORK A LITTLE HARDER.
--Your friends at LectureNotes
Close

Probability and Statistics

by Jntu Heroes
Type: NoteInstitute: Jawaharlal nehru technological university anantapur college of engineering Offline Downloads: 124Views: 2120Uploaded: 9 months agoAdd to Favourite

Share it with your friends

Suggested Materials

Leave your Comments

Contributors

Jntu Heroes
Jntu Heroes
Random Variable Motivation example In an opinion poll, we might decide to ask 50 people whether they agree or disagree with a certain issue. If we record a “1” for agree and “0” for disagree, the sample space for this experiment has 250 elements. If we define a variable X=number of 1s recorded out of 50, we have captured the essence of the problem. Note that the sample space of X is the set of integers {1, 2, . . . , 50} and is much easier to deal with than the original sample space. In defining the quantity X, we have defined a mapping (a function) from the original sample space to a new sample space, usually a set of real numbers. In general, we have the following definition. Definition of Random Variable A random variable is a function from a sample space S into the real numbers. Example 1.4.2 (Random variables) In some experiments random variables are implicitly used; some examples are these. Experiment Random variable Toss two dice X =sum of the numbers Toss a coin 25 times X =number of heads in 25 tosses Apply different amounts of fertilizer to corn plants X =yield/acre Suppose we have a sample space S = {s1 , . . . , sn } with a probability function P and we define a random variable X with range X = {x1 , . . . , xm }. We can define a probability function PX on X in the following way. We will observe X = xi if and only if the outcome of the random experiment is an sj ∈ S such that X(sj ) = xi . Thus, PX (X = xi ) = P ({sj ∈ S : X(sj ) = xi }). 1 (1)
Note PX is an induced probability function on X , defined in terms of the original function P . Later, we will simply write PX (X = xi ) = P (X = xi ). Fact The induced probability function defined in (1) defines a legitimate probability function in that it satisfies the Kolmogorov Axioms. Proof: CX is finite. Therefore B is the set of all subsets of X . We must verify each of the three properties of the axioms. (1) If A ∈ B then PX (A) = P (∪xi ∈A {sj ∈ S : X(sj ) = xi }) ≥ 0 since P is a probability function. (2) PX (X ) = P (∪m i=1 {sj ∈ S : X(sj ) = xi }) = P (S) = 1. (3) If A1 , A2 , . . . ∈ B and pairwise disjoint then ∞ PX (∪∞ k=1 Ak ) = P (∪k=1 {∪xi ∈Ak {sj ∈ S : X(sj ) = xi }}) = ∞ X P (∪xi ∈Ak {sj ∈ S : X(sj ) = xi } = ∞ X PX (Ak ), k=1 k=1 where the second inequality follows from the fact P is a probability function. ¤ A note on notation: Random variables will always be denoted with uppercase letters and the realized values of the variable will be denoted by the corresponding lowercase letters. Thus, the random variable X can take the value x. Example 1.4.3 (Three coin tosses-II) Consider again the experiment of tossing a fair coin three times independently. Define the random variable X to be the number of heads obtained in the three tosses. A complete enumeration of the value of X for each point in the sample space is s X(s) HHH HHT 3 HTH THH TTH THT HTT TTT 2 2 1 1 1 0 2 The range for the random variable X is X = {0, 1, 2, 3}. Assuming that all eight points in S have probability 81 , by simply counting in the above display we see that the induced probability function on X is given by 2
x 0 1 2 3 PX (X = x) 1 8 3 8 3 8 1 8 The previous illustrations had both a finite S and finite X , and the definition of PX was straightforward. Such is also the case if X is countable. If X is uncountable, we define the induced probability function, PX , in a manner similar to (1). For any set A ⊂ X , PX (X ∈ A) = P ({s ∈ S : X(s) ∈ A}). (2) This does define a legitimate probability function for which the Kolmogorov Axioms can be verified. Distribution Functions Definition of Distribution The cumulative distribution function (cdf) of a random variable X, denoted by FX (x), is defined by FX (x) = PX (X ≤ x), for all x. Example 1.5.2 (Tossing three coins) Consider the experiment of tossing three fair coins, and let X =number of heads observed. The    0       1    8   FX (x) = 1 2      7   8      1 cdf of X is if −∞ < x < 0 if 0 ≤ x < 1 if 1 ≤ x < 2 if 2 ≤ x < 3 if 3 ≤ x < ∞. 3
Remark: 1. FX is defined for all values of x, not just those in X = {0, 1, 2, 3}. Thus, for example, 7 FX (2.5) = P (X ≤ 2.5) = P (X = 0, 1, 2) = . 8 2. FX has jumps at the values of xi ∈ X and the size of the jump at xi is equal to P (X = xi ). 3. FX = 0 for x < 0 since X cannot be negative, and FX (x) = 1 for x ≥ 3 since x is certain to be less than or equal to such a value. FX is right-continuous, namely, the function is continuous when a point is approached from the right. The property of right-continuity is a consequence of the definition of the cdf. In contrast, if we had defined FX (x) = PX (X < x), FX would then be left-continuous. Theorem 1.5.3 The function FX (x) is a cdf if and only of the following three conditions hold: a. limx→−∞ F (x) = 0 and limx→∞ F (x) = 1. b. F (x) is a nondecreasing function of x. c. F (x) is right-continuous; that is, for every number x0 , limx↓x0 F (x) = F (x0 ). Example 1.5.4 (Tossing for a head) Suppose we do an experiment that consists of tossing a coin until a head appears. Let p =probability of a head on any given toss, and define X =number of tosses required to get a head. Then, for any x = 1, 2, . . ., P (X = x) = (1 − p)x−1 p. The cdf is FX (x) = P (X ≤ x) = x X P (X = i) = i=1 x X (1 − p)i−1 p = 1 − (1 − p)x . i=1 It is easy to show that if 0 < p < 1, then FX (x) satisfies the conditions of Theorem 1.5.3. First, lim FX (x) = 0 x→−∞ 4

Lecture Notes