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

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

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

Uniform random variable

Uniform random variable

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

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?

Uniform random variable

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

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

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

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

Normal random variable To prove is a distribution function:

Normal random variable

Normal random variable

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

Normal random variable

Normal random variable

Normal random variable Correction for continuity

Normal random variable Histogram of X and the density function f

Normal random variable

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?

Normal random variable Sol:

Normal random variable Continue:

Normal random variable

Normal random variable

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

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).

Normal random variable

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?

Normal random variable Sol:

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 100. What should the score of a student be to place him among the to 10% of all students?

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

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

Exponential random variable

Exponential random variable

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.

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?

Exponential random variable Sol:

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?

Exponential random variable Sol:

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

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

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:

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