Discussion about random variable ad its characterization

Discussion about random variable ad its characterization

1. Random variables and its Characterization Introduction

2. Random variable A numerical value, determined by chance, for each outcome of a procedure.

3. Random Variable Let us consider a random experiment of tossing three coins (or a coin is tossing three times). Then sample space is S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT} We are interested in the number of heads in each outcome. Let X denote the number of heads in each outcome, then X takes the values 0,1,2,3

4. Random Variable (Cont.) X (HHH) = 3, X(HHT) = 2, X(HTH) = 2, X(THH) = 2 , X(HTT) = 1, X (THT) = 1, X (TTH) = 1, X(TTT) = 0 i.e. X = 0, 1, 2 or 3 X is a variable obtained from the random experiment and is called random variable .

5. Random variable Types 5

6. Example: The number of eggs that a hen lays Another Example Discrete or Continuous

7. Example: The amount of milk that a cow produces; e.g. 2.343115 gallons per day Another Example Discrete or Continuous

8. Random Variables Informally, a random variable ( r.v. ) denotes possible outcomes of an event Can be discrete (i.e., finite many possible outcomes) or continuous Some examples of discrete r.v. denoting outcomes of a coin-toss denoting outcome of a dice roll Some examples of continuous r.v. denoting heights of students in MTH3 denoting time to get to your classroom from Unimall   8     (a discrete r.v. )   (a continuous r.v. )  

9. Probability of a random variable A probability is usually expressed in terms of a random variable. For the length of a manufactured part, if X denotes the length. Then the probability statement can be written in either of the following forms Both equations state the probability that the random variable X assumes a value in [10.8, 11.2] is 0.25.

10. Probability distribution A description that gives the probability for each value of the random variable; often expressed in the format of a graph, table, or formula.

11. Probability Distribution (Example revisited) Lets find the probability of each outcome in the example of getting is P(X = 0) = Probability of getting no head = P(TTT) = P(X = 1) = Probability of getting one head = P(HTT or THT or TTH ) = P(X = 2) = Probability of getting two heads = P(HHT or THH or HTH ) = P(X = 3) = Probability of getting three heads = P(HHH) =

12. Cont. Probability distribution of X is: P(X) is called the probability distribution of the random variable X.

13. Discrete Random Variables For a discrete r.v. , denotes = probability that is called the probability distribution or probabilty mass function (PMF) of r.v. or is the value of the PMF at   13      

14. Cumulative Distribution Function Let X be a random variable then Where P(X ≤ x )is the probability that X is less than or equal to x is called the cumulative distribution function ( d,f .) of X.

15. Properties of Cumulative Distribution Function.

16. Probability mass function ( pmf ) x p(x) 1 p(x=1) =1/6 2 p(x=2) =1/6 3 p(x=3) =1/6 4 p(x=4) =1/6 5 p(x=5) =1/6 6 p(x=6) =1/6 On rolling a dice, following are the probabilities:

17. Discrete example: roll of a die x p(x) 1/6 1 4 5 6 2 3

18. Cumulative distribution function x P(x≤A) 1 P(x≤1) =1/6 2 P(x≤2) =2/6 3 P(x≤3) =3/6 4 P(x≤4) =4/6 5 P(x≤5) =5/6 6 P(x≤6) =6/6

19. Cumulative distribution function (CDF) x P(x) 1/6 1 4 5 6 2 3 1/3 1/2 2/3 5/6 1.0

20. Practice Problem The number of patients seen in a clinic in any given hour is a random variable represented by x . The probability distribution for x is: x 10 11 12 13 14 P(x) 0.4 0.2 0.2 0.1 0.1 Find the probability that in a given hour: a.     exactly 14 patients arrive b.     At least 12 patients arrive c.     At most 11 patients arrive   p(x=14) = .1 p(x  12) = (.2 + .1 +.1) = .4 p(x≤11) = (.4 +.2) = .6

21. A word about notation 21 can mean different things depending on the context denotes the distribution (PMF/PDF) of an r.v. or or simply denotes the prob. or prob. density at value  

22. Probability distribution 22

23. Continuous Random Variables For a continuous r.v. , a probability or is meaningless For continuous r.v. , prob. Is discussed for an interval is the prob. that as is the probability density at    

24. For example, the probability of x falling within 1 to 2: x p(x)=e -x 1 1 2

25. Continuous random variables Probability density function

26. Continuous random variables Cumulative distribution function

27. s

28. Example x f(x) p(x) 1 0.2 0.2 2 0.32 0.12 3 0.67 0.35 4 0.9 0.23 5 1 0.1 Total= 1 Hence, P(x=3)=0.35 P(x>2)=1-p(x<=2) =1-(0.32) =0.68

29. Example Suppose  X  is a discrete random variable. Let the pmf of  X  be given by f(x)=(5-x)/10 for x=1,2,3,4 Find cdf of X

30. Ex From a lot of 10 items containing 3 defectives, a sample of 4 items is drawn at random. If X denotes the number of defective items in a sample. Solution: Let us consider 10 items as 1 2 3 4 5 6 7 8 9 10 Total number of outcomes if 4 items are selected, is  

31. 1 2 3 4 5 6 7 8 9 10 Let round marked items are defectives If X denotes the number of defectives, then X=0, 1, 2, 3; since total number of defective items are 3. Case 1: No defective Thus, the number outcomes having no defectives is= And P(no defective)= Case 2: 1 defective { 1 , { 2 , = { 3 , Thus, P(1 defective)=  

32. Case 3: 2 defectives { 1 , 2, { 1 , 3, = { 2, 3 , Thus, P(2 defectives)= Case 4: 3 defectives There are 7 such cases { 1,2,3 ,4}; { 1,2,3 ,5}; { 1,2,3 ,6}, { 1,2,3 ,7}, { 1,2,3 ,8}, { 1,2,3 ,9}, { 1,2,3 ,10} Thus, P(3 defectives) Thus probability distribution is   X 1 2 3 P(x)=f(x) 1/6 1/2 3/10 1/30

33. (ii) Now,  

34. Important Examples

35. Try by yourself Probability function of (ii)Y (iii) Z, (iv) X +Y and (v) XY

36. z

37. To draw Probability chart

38. Example The probability density function of a random variable x is given as: for Determine the value of the constant k. the distribution function F(x) ( 5/4  

39. 1

40. Example A random variable is exponentially distributed with probability distribution function as =0 Find  

41. Thank You...