1. Random Variables Introduction to Probability & Statistics Random Variables

2. Random Variables A Random Variable is a function that associates a real number with each element in a sample space. Ex: Toss of a die X = # dots on top face of die = 1, 2, 3, 4, 5, 6

3. Random Variables A Random Variable is a function that associates a real number with each element in a sample space. Ex: Flip of a coin 0, heads X = 1, tails 

4. Random Variables A Random Variable is a function that associates a real number with each element in a sample space. Ex: Flip 3 coins 0 if TTT X = 1 if HTT, THT, TTH 2 if HHT, HTH, THH 3 if HHH 

5. Random Variables A Random Variable is a function that associates a real number with each element in a sample space. Ex: X = lifetime of a light bulb X = [0,  )

6. Distributions Let X= number of dots on top face of a die when thrown p(x) = Prob{X=x} x 1 2 3 4 5 6 p(x) 1 / 6 1 / 6 1 / 6 1 / 6 1 / 6 1 / 6

7. Cumulative Let F(x) = Pr{X < x} x 1 2 3 4 5 6 p(x) 1 / 6 1 / 6 1 / 6 1 / 6 1 / 6 1 / 6 F(x) 1 / 6 2 / 6 3 / 6 4 / 6 5 / 6 6 / 6

8. Complementary Cumulative Let F(x) = 1 - F(x) = Pr{X > x} x 1 2 3 4 5 6 p(x) 1 / 6 1 / 6 1 / 6 1 / 6 1 / 6 1 / 6 F(x) 1 / 6 2 / 6 3 / 6 4 / 6 5 / 6 6 / 6 F(x) 5 / 6 4 / 6 3 / 6 2 / 6 1 / 6 0 / 6

9. Discrete Univariate u Binomial u Discrete Uniform (Die) u Hypergeometric u Poisson u Bernoulli u Geometric u Negative Binomial

10. Binomial Distribution 0.0 0.1 0.2 0.3 0.4 0.5 012345 x P(x) 0.0 0.1 0.2 0.3 0.4 0.5 012345 x P(x) n=5, p=.3 n=8, p=.5 x 0.0 0.1 0.2 0.3 0.4 0.5 012345678 P(x) n=4, p=.8 0.0 0.1 0.2 0.3 0.4 0.5 024 x P(x) n=20, p=.5

11. Binomial Measures Mean : Variance:   xpx x ()  22   ()()xpx x = np = np(1-p)

12. Continuous Distribution x a b c d f(x) A 1.f(x) > 0, all x 2. 3.P(A) = Pr{a < x < b} = 4.Pr{X=a} = fxdx a d ()   1 fx b c ()  fx a a ()   0

13. Continuous Univariate u Normal u Uniform u Exponential u Weibull u LogNormal u Beta u T-distribution u Chi-square u F-distribution u Maxwell u Raleigh u Triangular u Generalized Gamma u H-function

14. Normal Distribution  65% 95% 99.7%

15. Std. Normal Transformation  Standard Normal Z X    f(z) N(0,1)

16. Example u Suppose a resistor has specifications of 100 + 10 ohms. R = actual resistance of a resistor and R N(100,5). What is the probability a resistor taken at random is out of spec? x LSLUSL 100  

17. Example Cont. x LSLUSL 100  Pr{in spec}= Pr{90 < x < 110}             Pr 90100 5 110100 5 x   = Pr(-2 < z < 2)

18. Example Cont. x LSLUSL 100  Pr{in spec} = Pr(-2 < z < 2) = [F(2) - F(-2)] = (.9773 -.0228) =.9545

19. Example Cont. x LSLUSL 100  Pr{in spec} = Pr(-2 < z < 2) = [F(2) - F(-2)] = (.9773 -.0228) =.9545 Pr{out of spec} = 1 - Pr{in spec} = 1 -.9545 = 0.0455

20. Example Suppose the distribution of student grades for university are approximately normally distributed with a mean of 3.0 and a standard deviation of 0.3. What percentage of students will graduate magna or summa cum laude? x 3.0 

21. Example Cont. Pr{magna or summa} = Pr{X > 3.5}} = Pr(z > 1.67) = 0.5 - 0.4525 = 0.0475 x 3.0            3.0 0.35.3 Pr   X

22. Example u Suppose we wish to relax the criteria so that 10% of the student body graduates magna or summa cum laude. x 3.0  0.1

23. Example u Suppose we wish to relax the criteria so that 10% of the student body graduates magna or summa cum laude. x 3.0  0.1 0.1 = Pr{Z > z} z = 1.282

24. Example But x 3.0  0.1    X Z x =  +  z = 3.0 + 0.3 x 1.282 = 3.3846

25. Exponential Distribution fxe x ()   Density Cumulative Mean 1/ Variance 1/ 2 Fxe x ()   1, x > 0 0.0 0.5 1.0 00.511.522.53 Time to Fail Density =1

26. Exponential Distribution fxe x ()   Density Cumulative Mean 1/ Variance 1/ 2 Fxe x ()   1, x > 0 =1 0.0 0.5 1.0 1.5 2.0 00.511.522.53 Time to Fail Density =2

27. Example Let X = lifetime of a machine where the life is governed by the exponential distribution. determine the probability that the machine fails within a given time period a., x > 0, > 0 fxe x ()  

28. Example Exponential Life 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 00.511.522.53 Time to Fail Density a fxe x ()   f(x) FaXa()Pr{}     edx x a 0   e xa 0

29. Example Exponential Life 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 00.511.522.53 Time to Fail Density a fxe x ()   f(x) FaXa()Pr{}     edx x a 0   e xa 0   1e a

30. Example fxe x ()   Exponential Life 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 00.511.522.53 Time to Fail Density a f(x) FaXa()Pr{}     edx x a 0   e xa 0   1e a Note: F(  ) = 1-e -  = 1 F(0) = 1 - e - 0 = 0

31. Complementary Suppose we wish to know the probability that the machine will last at least a hrs? Exponential Life 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 00.511.522.53 Time to Fail Density a f(x) FaXa()Pr{}     edx x  a   e a

32. Example Suppose for the same exponential distribution, we know the probability that the machine will last at least a more hrs given that it has already lasted c hrs. Pr{X > a + c | X > c}= Pr{X > a + c  X > c} / Pr{X > c} = Pr{X > a + c} / Pr{X > c} c a c+a     e e e ca c a ()