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 Birth of a Distribution Class Width = 10 Cyberland Wages

3 Birth of a Distribution Class Width = 5

4 Birth of a Distribution Class Width = 2

5 Birth of a Distribution Class Width = 1

6 Birth of a Distribution Class Width = Very Small

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

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

9 Normal Distributions  = 5  =2  =3  =

10 Normal Distributions Same , Different 

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

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

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

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

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

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

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*) = Find z such that P(z > z ) = Find t such that P(t > t ) = with 5 degrees of freedom

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 z  /2 for  = z Standard Normal Distribution (z) z P(z < ) = P(z > ) = P( < z < ) = 0.95 ??

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

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

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

23  2 Distribution P(  2 > ) = 0.05 P(  2 < ) = d.f

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

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

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

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 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?