1. Part VI: Named Continuous Random Variables

2. Comparison of Named Distributions
discrete
continuous
Bernoulli, Binomial, Geometric, Negative Binomial, Poisson, Hypergeometric, Discrete Uniform
Continuous Uniform, Exponential, Gamma, Beta, Normal

3. Chapter 30: Continuous Uniform R.V.

4. Uniform distribution: Summary
Things to look for: constant density on a line or area Variable: X = an exact position or arrival time Parameter: (a,b): the endpoints where the density is nonzero. Density: CDF: 𝑓 𝑥 𝑥 = 1 𝑏−𝑎 𝑎≤𝑥≤𝑏 0 𝑒𝑙𝑠𝑒 𝐹 𝑋 𝑥 = 0 𝑥<𝑎 𝑥−𝑎 𝑏−𝑎 𝑎≤𝑥≤𝑏 1 𝑏<𝑥 𝔼 𝑋 = 𝑎+𝑏 2 , 𝑉𝑎𝑟 𝑋 = (𝑏−𝑎) 2 12

5. Example: Uniform Distribution (Class)
A bus arrives punctually at a bus stop every thirty minutes. Each morning, a bus rider leaves her house and casually strolls to the bus stop.
Why is this a Continuous Uniform distribution situation? What are the parameters? What is X?
What is the density for the wait time in minutes?
What is the CDF for the wait time in minutes?
Graph the density.
Graph the CDF.
What is the expected wait time?

6. Example: Uniform Distribution (Class)
A bus arrives punctually at a bus stop every thirty minutes. Each morning, a bus rider leaves her house and casually strolls to the bus stop.
What is the standard deviation for the wait time?
What is the probability that the person will wait between 20 and 40 minutes? (Do this via 3 different methods.)
Given that the person waits at least 15 minutes, what is the probability that the person will wait at least 20 minutes?

7. Example: Uniform Distribution

8. Example: Uniform Distribution (Class)
A bus arrives punctually at a bus stop every thirty minutes. Each morning, a bus rider leaves her house and casually strolls to the bus stop.
Let the cost of this waiting be $20 per minute plus an additional $5.
What are the parameters?
What is the density for the cost in minutes?
What is the CDF for the cost in minutes?
What is the expected cost to the rider?
What is the standard deviation of the cost to the rider?

9. Chapter 31: Exponential R.V.

10. Exponential Distribution: Summary
Things to look for: waiting time until first event occurs or time between events. Variable: X = time until the next event occurs, X ≥ 0 Parameter: : the average rate Density: CDF: 𝑓 𝑥 𝑥 = 𝜆 𝑒 −𝜆𝑥 𝑥>0 0 𝑒𝑙𝑠𝑒 𝐹 𝑋 𝑥 = 1− 𝑒 −𝜆𝑥 𝑥>0 0 𝑒𝑙𝑠𝑒 𝔼 𝑋 = 1 𝜆 , 𝑉𝑎𝑟 𝑋 = 1 𝜆 2

11. Example: Exponential R.V. (class)
Suppose that the arrival time (on average) of a large earthquake in Tokyo occurs with an exponential distribution with an average of 8.25 years.
What does X represent in this story? What values can X take?
Why is this an example of the Exponential distribution?
What is the parameter for this distribution?
What is the density?
What is the CDF?
What is the standard deviation for the next earthquake?

12. Example: Exponential R.V. (class, cont.)
Suppose that the arrival time (on average) of a large earthquake in Tokyo occurs with an exponential distribution with an average of 8.25 years.
What is the probability that the next earthquake occurs after three but before eight years?
What is the probability that the next earthquake occurs before 15 years?
What is the probability that the next earthquake occurs after 10 years?
How long would you have to wait until there is a 95% chance that the next earthquake will happen?

13. Example: Exponential R.V. (Class, cont.)
Suppose that the arrival time (on average) of a large earthquake in Tokyo occurs with an exponential distribution with an average of 8.25 years. k) Given that there has been no large Earthquakes in Tokyo for more than 5 years, what is the chance that there will be a large Earthquake in Tokyo in more than 15 years? (Do this problem using the memoryless property and the definition of conditional probabilities.)

14. Minimum of Two (or More) Exponential Random Variables
Theorem 31.5 If X1, …, Xn are independent exponential random variables with parameters 1, …, n then Z = min(X1, …, Xn) is an exponential random variable with parameter 1 + … + n.

15. Chapter 37: Normal R.V.

16. Normal Distribution: Summary
Things to look for: bell curve, Variable: X = the event Parameters: X = the mean 𝜎 𝑋 2 =𝑡ℎ𝑒 𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 Density: 𝑓 𝑥 𝑥 = 𝑒 − (𝑥− 𝜇 𝑥 ) 2 (2 𝜎 2 ) 2𝜋 𝜎 2 , 𝑥 𝜖 ℝ 𝔼 𝑋 =X, 𝑉𝑎𝑟 𝑋 = 𝜎 𝑋 2

17. PDF of Normal Distribution (cont)

18. PDF of Normal Distribution

20. PDF of Normal Distribution (cont)

21. Procedure for doing Normal Calculations
Sketch the problem.
Write down the probability of interest in terms of the original problem.
Convert to standard normal.
Convert to CDFs.
Use the z-table to write down the values of the CDFs.
Calculate the answer.

22. Example: Normal r.v. (Class)
The gestation periods of women are normally distributed with  = 266 days and  = 16 days. Determine the probability that a gestation period is
less than 225 days.
between 265 and 295 days.
more than 276 days.
less than 300 days.
Among women with a longer than average gestation, what is the probability that they give birth longer than 300 days?

23. Example: “Backwards” Normal r.v. (Class)
The gestation periods of women are normally distributed with  = 266 days and  = 16 days. Find the gestation length for the following situations:
longest 6%.
shortest 13%.
middle 50%.

24. Example: Normal r.v. (class)
Suppose that a Scottish soldier's chest size is normally distributed with  = 39.8 inches and  = 2.05 inches. a) What is the probability that a Scottish soldier has a chest size of less than 35 inches? b) What is the probability that a Scottish soldier has a chest of at least 40 inches? c) What is the probability that a Scottish soldier has a chest size between 35 and 40 inches?

25. Chapter 35: Sums of Independent Normal Random Variables

26. Chapter 36: Central Limit Theorem (Normal Approximations to Discrete Distributions – 36.4, 36.5)
/normal.html

27. Continuity Correction - 1

28. Continuity Correction - 2
W~N(10, 5)
X ~ Binomial(20, 0.5)

29. Continuity Correction - 3
Discrete
Continuous
a < X
a < X
a ≤ X
a – 0.5 < X
X < b
X < b – 0.5
X ≤ b
X < b + 0.5

30. Normal Approximation to Binomial

31. Example: Normal Approximation to Binomial (Class)
The ideal size of a first-year class at a particular college is 150 students. The college, knowing from past experience that on the average only 30 percent of these accepted for admission will actually attend, uses a policy of approving the applications of 450 students.
Compute the probability that more than 150 students attend this college.
Compute the probability that fewer than 130 students attend this college.

32. Chapter 32: Gamma R.V.
/process_simulations_sensitivity_analysis_and_error_analysis_modeling
/distributions_for_assigning_random_values.htm

33. Gamma Distribution Generalization of the exponential function Uses
probability theory
theoretical statistics
actuarial science
operations research
engineering

34. Gamma Function
(t + 1) = t (t), t > 0, t real (n + 1) = n!, n > 0, n integer

35. Gamma Distribution: Summary
Things to look for: waiting time until rth event occurs Variable: X = time until the rth event occurs, X ≥ 0 Parameters: r: total number of arrivals/events that you are waiting for : the average rate Density: 𝑓 𝑥 𝑥 = 𝜆 𝑟 Γ(𝑟) 𝑥 𝑟−1 𝑒 −𝜆𝑥 𝑥>0 0 𝑒𝑙𝑠𝑒 𝔼 𝑋 = 𝑟 𝜆 , 𝑉𝑎𝑟 𝑋 = 𝑟 𝜆 2 𝐶𝐷𝐹: 𝐹 𝑋 𝑥 = 1− 𝑒 −𝜆𝑥 𝑗=0 𝑟−1 (𝜆𝑥) 𝑗 𝑗! 𝑥>0 0 𝑒𝑙𝑠𝑒

36. Gamma Random Variable k = r 𝜃= 1 𝜆

37. Chapter 33: Beta R.V.

38. Beta Distribution This distribution is only defined on an interval
standard beta is on the interval [0,1]
The formula in the book is for the standard beta
uses
modeling proportions
percentages
probabilities

39. Beta Distribution: Summary
Things to look for: percentage, proportion, probability Variable: X = percentage, proportion, probability of interest (standard Beta) Parameters: ,  Density: 𝑓 𝑥 𝑥 = 1 𝐵−𝐴 Γ(𝛼+𝛽) Γ(𝛼)Γ(𝛽) 𝑥−𝐴 𝐵−𝐴 𝛼−1 𝑥−𝐴 𝐵−𝐴 𝛽−1 𝐴≤𝑥≤𝐵 0 𝑒𝑙𝑠𝑒 Density: no simple form When A = 0, B = 1 (Standard Beta) 𝔼 𝑋 = 𝛼 𝛼+𝛽 , 𝑉𝑎𝑟 𝑋 = 𝛼𝛽 𝛼+𝛽 2 (𝛼+𝛽+1)

40. Shapes of Beta Distribution X

41. Other Continuous Random Variables
Weibull
exponential is a member of family
uses: lifetimes
lognormal
log of the normal distribution
uses: products of distributions
Cauchy
symmetrical, flatter than normal

42. Chapter 37: Summary and Review of Named Continuous R.V.
/univariate-continuous-distributions.html

43. Summary of Continuous Distributions
Name
Density, fX(x)
Domain
CDF, FX(x)
𝔼(X)
Var(X)
Parameters
What X is
When used

44. Expected values and Variances for selected families of continuous random variables.
Family
Parameter(s)
Expectation
Variance
Uniform
a,b
Exponential l
Normal
m,s2 m 2