Lecture 6 Random Variable. Compose of procedure & observation. From observation, we get outcomes

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1 ENM 07 Lecture 6 Random Variable Random Variable Eperiment (hysical Model) Compose of procedure & observation From observation we get outcomes From all outcomes we get a (mathematical) probability model called Sample space From the model we get [A] A S

Random Variable Definition: If ε is an eperiment having sample space S and is a function that assigns a real number (e) to every outcome e; e S then (e) is called a random variable S R e (e) (a) (b)

2 Random Variable Definition: If ε is an eperiment having sample space S and is a function that assigns a real number (e) to every outcome e; e S then (e) is called a random variable S R e (e) (a) (b) (a) The sample space of ε. (b) R : The range of space. Eample: Lets consider the coin tossing eperiment that true coin was tossed three times and the sample space is S{HHHHHTHTHHTTTHHTHTTTHTTT} If is the number of heads showing then: ( HHH ) 3 ( HHT ) ( HTH ) ( HTT ) ( THH ) ( THT ) ( TTH ) ( TTT ) 0 (They can be represented by. Same random variables.)

3 The range space R{:03} in this eample S R TTT. TTH. THT. HTT. HHT. HTH. THH. HHH If the number of heads showing in any outcome is same in the different outcome as you see in this figure different values of e may lead to the same. The range space R is made up of the possible values of. In coin tossing eperiment if the coin is true then there are eight equally likely outcomes each having probability /8. Lets A is the event eactly two heads represent the number of heads. The event that () relates to R not S: A{ HHTHTHTHH } (therefore) ( )(A)3/8 (eight equally likely outcomes in S)

4 Random Variable Definition: If S is the sample space of an eperiment ε and a random variable with range space R is defined on S and if event A is an event in S while event B is an event in R than A and B are equivalent events if A{e S: (e) B } Definition: If A is an event in the sample space S and B is an event in the range space R of the random variable then we define the probability of B as (B)(A) where A{e S:(e) B} The inverse of the function : ( B) { e S : ( e) B} ( ) ( A) ( B) ( B) Eample: Consider the tossing of two true dice Lets Y as the sum of the up faces. Then Ry { } probabilities are There are outcomes which since the dice are true are equally likely : (); 3: ()-(); 4: ()-(3)-(3); 5: (3)-(3)-(4)-(4); 6: (33)-(4)-(4)-(5)-(5); 7: (34)-(43)-(5)-(5)-(6)-(6)

5 Random Variable Discrete Random Variables Eamples: # of customers in a bank Y sum of the up faces in a rolling two dice Continuous Random Variables Eamples: processing time for a product Y height of a person Discrete Random Variable Definition: is a discrete random variable if the range of is countable S { } is a finite random variable if all values with nonzero probability are in the finite set S { n}

6 Discrete robability Distributions For a (discrete) probability model [A] [0] For a discrete random variable the probability model is called a robability Mass Function (MF) or robability Distribution Discrete robability Distributions

7 Discrete robability Distributions Definition: If is a discrete random variable it can be associated a number ( i ) p( i ) with each outcome i in R for i n where.for. p( ) R any p (B) ( ) 0 3.For any event B R that is in the set B is i p( ) B for all i ( B) the probability Eample: In the coin tossing eperiment where is the number of heads. We represent the probability distribution as a tabular or graphical form. i) Tabular Representation ii ) Graphical Representation () p() 0 /8 3/8 3/8 3/8 /8 3 / r.v. represents the outcome that it doesn t contain head represents the outcome that it contains one head represents two heads 3 represents three heads.

8 Eample: 3.3 (Walpole et al. p 66 ) A shipment of 8 similar microcomputers to a retail outlet contains 3 that are defective. If a school makes a random purchase of of these computers find the probability distribution for the number of defectives. Eample There are 5 firing pins of which 3 are defective. Let Y number of firing pins tested until the first defective is found. Let Di represents defective firing pin on the ith trial and Let Gi represents a good firing pin on the ith trial. Define probability distribution function for Y

9 Continuous Random Variables In Discrete: countable set of numbers R {-034} In Continuous: uncountable set of numbers R Interval between limits R () (-3) Continuous Random Variables Measuring T the download time R T {t 0 < t < } Guess the download time is (0 0] minutes Guess the download time is [5 8] minutes Guess the download time is [5 5.5] minutes Chance that our guess is correct is decreasing Guess the download time is eactly 5.5 min. robability of each individual outcome is zero. The interesting probability is an interval.

10 Continuous robability Distributions Suppose that is a random variable its sample space is S whose random variable function set (S) is a continuum of numbers such as an interval. The set B{ a b } is an event in S and therefore the probability (B)(a b) is defined. There is a piecewise continuous function. f : R R such that (a b) is equal to the area under the graph of a and b. In the language of calculus ( a b ) f ( ) In this case said to be a continuous random variable. b a d ( a b) the area of yellow shaded region The function f is called the distribution or the density function of. It satisfies the conditions i ii ) f ( ) ) f ( ) d R 0 Where R represents the real numbers set. That is f is nonnegative and the total area under its graph is. This area is equal to the probability of S.

11 Continuous robability Distributions Eample: Let be a continuous random variable with the following distribution : if 0 f ( ) 0 o. w. 5 5 ( ) ( 5 ) () 5 5 d As you see the density function f() is the simple linear function. The area of shaded region in this diagram is a trapezoid. You can use geometrical relation to find the same probability value ( 5) Continuous robability Distributions Theorem: The function f() is a probability density function for continuous random variable defined over the set of real numbers R if f f ( ) f ( ) 0 R d It is piecewise for all R continuous ( ) 0 if is not in the range R

12 Eample: 3-3/p.74 Montgomery λ t λ e t 0 f ( ) 0 o. w. λ t 00λ T λ e dt e ( 00) 00 ( T 00 \ T > 99 ) ( T 00 and T > 99 ) T ( T > 99 ) λ e λ t dt 00 λ t λ e dt 99 ( T 00 ) ( T > 99 ) 00 λ e λ e 99 λ e Eample: 3-4/p74. Montgomery f ( ) 0 0 < < o. w. f() The r.v. betveen 0 and belongs to the first function of and r.v. between and belongs to the second function. f () f ()- 0

13 ) ) ) ( ) ) ( ) )? 3 4? 5? 3? 3? < < < < e d c b a References Walpole Myers Myers Ye (00) robability & Statistics for Engineers & Scientists Dengiz B. (004) Lecture Notes on robability Hines Montgomery (990) robability & Statistics in Engineering & Management Science

Mathematics. ( : Focus on free Education) (Chapter 16) (Probability) (Class XI) Exercise 16.2

Mathematics. ( : Focus on free Education) (Chapter 16) (Probability) (Class XI) Exercise 16.2 ( : Focus on free Education) Exercise 16.2 Question 1: A die is rolled. Let E be the event die shows 4 and F be the event die shows even number. Are E and F mutually exclusive? Answer 1: When a die is