1. 1 Probability Distributions Random Variable A numerical outcome of a random experiment Can be discrete or continuous Generically, x Probability Distribution The pattern of probabilities associated with all of the random variables for a specific experiment Can be a table, formula, or graph Generically, f(x) Examples Binomial (but won’t cover here) Uniform Normal or bell-shaped distribution

2. 2 Birth of a Distribution Class Width = 10 Cyberland Wages

3. 3 Birth of a Distribution Class Width = 5

4. 4 Birth of a Distribution Class Width = 2

5. 5 Birth of a Distribution Class Width = 1

6. 6 Birth of a Distribution Class Width = Very Small

7. 7 Uniform Distribution x f(x) 1 / (b-a) ab Area = 1

8. 8 Normal Distribution Bell-shaped, symmetrical distribution f(x) x

9. 9 Normal Distributions  = 5  =2  =3  =1 -2 12

10. 10 Normal Distributions Same , Different 

11. 11 Normal Distributions  68.26% ++--

12. 12 Normal Distributions  95.44%  +2  -2 

13. 13 Normal Distributions  99.72%  +3  -3 

14. 14 Standard Normal Distribution z 0  z = 1  z = 0 If x has a normal distribution…

15. 15 t Distribution 3.5 0 but has thicker tails -3.5 Specific thickness depends on degrees of freedom Looks like a normal distribution,

16. 16 t Distribution 3.5 0 -3.5 Specific thickness depends on degrees of freedom 5 d.f. 10 d.f. 30 d.f. 100 d.f.  d.f (normal)

17. 17 Find the Probabilities 1.P(z > 2.36) 2.P(t > -1.02) with 5 degrees of freedom 3.P(-0.95 < z < 1.93) 4.P(-0.95 < t < -0.07) with 100 degrees of freedom 5.Find z* such that P(z < z*) = 0.719 6.Find z 0.025 such that P(z > z 0.025 ) = 0.025 7.Find t 0.025 such that P(t > t 0.025 ) = 0.025 with 5 degrees of freedom

18. 18 z  /2 0 -z  /2 Standard Normal Distribution (z) z  /2 P(z < -z  /2) ) =  /2 P(z > z  /2 ) =  /2 P(-z  /2 < z < z  /2 ) = 1 -  /2

19. 19 z  /2 for  =0.05 0 -z 0.025 Standard Normal Distribution (z) z 0.025 P(z < ) = 0.025 P(z > ) = 0.025 P( < z < ) = 0.95 ??

20. 20 t  /2 for  =0.05, df=5 0 -t 0.025 t distribution with 5 degrees of freedom t 0.025 P(t < ) = 0.025 P(t > ) = 0.025 P( < t < ) = 0.95 ??

21. 21  2 Distribution 0 Specific skewness depends on degrees of freedom

22. 22  2 Distribution 0 Specific skewness depends on degrees of freedom 5 d.f 10 d.f 15 d.f

23. 23  2 Distribution 0 18.307 P(  2 > 18.307) = 0.05 P(  2 < 18.307) = 0.95 10 d.f

24. 24 F Distribution 0 Specific skewness depends on a pair of degrees of freedom (df1, df2)

25. 25 F Distribution 0 3.02 P(F > 3.02) = 0.05 P(F < 3.02) = 0.95 9 and 10 d.f

26. 26 Probability Distributions Different shapes and df’s, but SAME LOGIC ! Normal & t 22 F

27. 27 In Excel To find probability above a value x =1-NORMSDIST(x) =TDIST(x,df,1) [1=1-tail] =CHIDIST(x,df) =FDIST(x,df1,df2) To find value with p% above (e.g., 0.05) =NORMSINV(p) =TINV(p,df) =CHIINV(p,df) =FINV(p,df1,df2)

28. 28 Word Problem From past experience, the management of a well- known fast food restaurant estimates that the number of weekly customers at a particular location is normally distributed, with a mean of 5000 and a standard deviation of 800 customers. What is the probability that on a given week the number of customers will be between 4760 and 5800? What is the probability of a week with more than 6500 customers? For 90% of the weeks, the number of customers should exceed what amount?