1. Chap 7 Special Continuous Distributions Ghahramani 3rd edition
2019/2/24

2. Outline 7.1 Uniform random variable 7.2 Normal random variable
7.3 Exponential random variables
7.4 Gamma distribution
7.5 Beta distribution
7.6 Survival analysis and hazard function

3. 7.1 Uniform random variable
Def A random variable X is said to be uniformly distributed over an interval
(a, b) (written as X~U(a,b) in short) if its density function is

4. Uniform random variable

5. Uniform random variable

6. Uniform random variable
Comparison:
If Y is a discrete random variable selected
from the set { 1, 2, …, N }, then

7. Uniform random variable
Ex 7.3 What is the probability that a random chord of a circle is longer than a side of an equilateral triangle inscribed into the circle?

8. Uniform random variable

9. Uniform random variable
Sol:
(a)interpretation 1: P(d<r/2)=1/2
(b)interpretation 2: 1/3
(c)interpretation 3:

10. 7.2 Normal random variable
De Moivre’s Thm Let X~B(n,1/2) then for
a and b, a < b
Note that EX=n/2 and s.d.(X)=n1/2/2

11. Normal random variable
Thm 7.1 (De Moivre-Laplace Thm)
Let X~B(n,p) then for a and b, a < b
Note that EX=np and s.d.(X)=(np(1-p))1/2

12. Normal random variable
Def A random variable X is called standard normal (written as X~N(0,1)) if its distribution function is

13. Normal random variable
To prove is a distribution function:

14. Normal random variable

15. Normal random variable

16. Normal random variable
By the fundamental theorem of calculus, the density function f is
which is a bell-shaped curve that is symmetric about the y-axis

17. Normal random variable

18. Normal random variable

19. Normal random variable
Correction for continuity

20. Normal random variable
Histogram of X and the density function f

21. Normal random variable

22. Normal random variable
Ex 7.4 Suppose that of all the clouds that are seeded with silver iodide, 58% show splendid growth. If 60 clouds are seeded with silver iodide, what is the probability that exactly 35 show splendid growth?

23. Normal random variable
Sol:

24. Normal random variable
Continue:

25. Normal random variable

26. Normal random variable

27. Normal random variable
Def A random variable X is called normal, with parameters and (written as X~N( , )), if its density function is

28. Normal random variable
Lemma If X~N( , ), then Z=(X- )/ is N(0,1). That is , if X ~N( , ), the standardized X is N(0,1).

29. Normal random variable

30. Normal random variable
Ex 7.5 Suppose that a Scottish soldier’s chest size is normally distributed with mean 39.8 and standard deviation 2.05 inches, respectively. What is the probability that of 20 randomly selected Scottish soldiers, 5 have a chest of at least 40 inches?

31. Normal random variable
Sol:

32. Normal random variable
Ex 7.7 The scores on an achievement test given to 100,000 students are normally distributed with mean 500 and standard deviation What should the score of a student be to place him among the to 10% of all students?

33. Normal random variable
Sol: to find x such that P(X<x)=0.90.

34. 7.3 Exponential random variable
Def A continuous random variable X is called exponential with parameter >0 (written as X~EP( )) if its density function and distribution function are

35. Exponential random variable

36. Exponential random variable

37. Exponential random variable
Examples:
The interarrival time between 2 customers at a post office.
The duration of Jim’s next telephone call.
The time between 2 consecutive earthquakes in California.
The time between two accidents at an intersection.
The time until the next baby is born in a hospital.
The time until the next crime in a certain town.

38. Exponential random variable
Ex 7.10 Suppose that every 3 months, on average, an earthquake occurs in California. What is the probability that the next earthquake occurs after 3 but before 7 months?

39. Exponential random variable
Sol:

40. Exponential random variable
Ex 7.11 At an intersection, there are 2 accidents per day, on average. What is the probability that after the next accident there will be no accidents at all for the next 2 days?

41. Exponential random variable
Sol:

42. Exponential random variable
An important feature of exponential distribution is its memoryless property.

43. Exponential random variable
Exponential random variables are memoryless.
<proof>

44. Exponential random variable
Ex 7.12 The lifetime of a TV tube (in years) is an exponential random variable with mean 10. If Jim bought his TV set 10 years ago, what is the probability that its tube will last another 10 years?
Sol:

45. Skip
7.4 Gamma distribution and
7.5 Beta distribution
7.6 Survival analysis and hazard functions