Chapter 4 Continuous Distribution ( การแจกแจงความน่าจะเป็นแบบต่อเนื่อง)

Chapter 4

Continuous Distribution

Definition Continuous Random Variable

A random variable is continuous if it can assume any value in some interval or intervals of real numbers and the probability that it assume any specific value is 0.

Note 1.Owing to the limits of our ability to measure, experimental data never seem to come from a continuous scale.

Ex. T=74.8C is not a point on acontinuous scale. Actually it could be any value between74.75-74.85

2. Sincethe probability that a continuous random variable will take on a point on a scale is 0, therefore there is no difference in writing intervals.

P (a  x  b ) = P (a  x < b)
= P (a < x b)
= P(a < x <b)

Probability Distribution Function

Definition Let x be a continuous random variable. A function f is called probability density function if:
(1) f(x)  0 for x real
(2)
(3) P (a  x  b) = for a and b real

Ex.1 If a random variable has the probability density

12.5x -1.25 0.1  x  0.5

f(x) =

0 ,elsewhere

Find P (0.2  x  0.3)

Def. Cumulative Distribution Function
Let x be continuous with density f, the cumulative distribution function for x,
denoted by F, is defined by:
F(x) = P( X  x ) for x real

Computationally ,

P( X  x ) = F (x)
=

Determining probability from F(x)

1. P (X  x) = 1 - F(x)

2. P ( x1 X  x2 ) = F (x2 ) – F ( x1)

Properties of F (x)

F(x) is nondecreasing as x increases, meaning that

if x1 < x2 then F (x1) F(x2)

2. = 0 and = 1

Ex.2 Suppose the pdf of the magnitude X of a dynamic load on a bridge (in newtons) is

f (x) = , 0 x 2

0 otherwise

(a) Determine F(x)

(b) Find the prob. that the load is between 1 and 1.5 newtons

Expectation

The expected value (mean value) of a continuous random variable with pdf f(x) is
= E (X) =
If X is a continuous random variable with pdf f(x) and h(x) is any function of X, then
E [h(X)] = =

Ex. The distribution of the amount of gravel (in tons) sold by a particular construction supply company in a given week is a continuous random variable X with pdf

f(x) = , 0 x 1

0otherwise

( a) Find F(x) (b) Find E (x)

Variance

Definition The varianace of a continuous random variable X with pdf f(x) and mean value  is
= V(x) =E =
The standard deviation of X is
=

Shortcut formula V(x) = E (x2) - [E(x)]2

Ex. For X = weekly gravel sales, we computed E(X) = 3/8. Find V(X) and (x)

Rules for Variancesame as discrete case

------midterm------

Normal Distribution N (,2) or f (x; ,2)

underlies many of the statistical methods

is the limiting form of binomial density as the number of trials become infinite

describes the behavior of errors inmeasurements

Definition A random variable x has a normal distribution with parameters and if its density is given by :
f (x) =
, - x 
E (X) = 
Var X = 2

Graph of N (,2)

Moment Generating Function

Definition Let x be a random variable with density f . The moment generating function for x (m.g.f) is denoted by mx(t) and is given by:
= E (etx) for t real
Theorem Let be the moment generating functionfor x, then
= E [x k]

Standard Normal Distribution Table

Standard Normal Distribution หมายถึงnormal distribution ที่มี =0,  = 1

F (Z) =

Ex. Find the probabilities that a random variable having the standard normal distribution will take on a value

(a)between 0.87 and 1.28

(b) between -0.34 and 0.62

(d) greater than –0.65

In finding theprobability of random variable x withmeanand variance 2, we can use the standard normal table after changing x to z (normal standard normal)

Z =

Thus

P (a< x <b) = -

Ex. Let x = number of grams of hydrocarbons emitted by an automobile per mile. Assuming that x is normal with =1 g and  = 0.25g, find the probability that a randomly selected automobile will emit between 0.9 and 1.54 g of HC per mile

Ex. Let x = the amount of radiation that can be absorbed by an individual before death ensues. Assume thatx is normal with a mean of 500roentgensand  of 150 roentgens. Above what dosage level will only 5 % of those exposed survive?

The Normal Approximations to the Bionomial Distribution

Theorem Let x be binomial with parameters n and p. For large n, x is approximately normal with mean np and variance np(1-p).

Acceptable if
p 0.5 and np > 5 Or
p > 0.5 and n(1-p) > 5

Ex. If 20 % of the diodes made in a certain plant are defective, what are the prob. that in a lot of 100 randomly chosen for inspection.

(a)at most 15 will be defective.

(b) exactly 15 will be defective ?

Linear Combinations of Normal Distributed Variable

Theorem1 If the random variables X1 ,…, Xk are independent and if Xihas a normal distribution with mean i and variance (i = 1, …, k), then the sum X1 + … + Xk

has a normal distribution with mean 1 + … + k and variance +…+

Theorem 2 If x has a normal distribution with mean and variance 2 and if y = ax + b, where a and b are constants and a 0, then Y has a normal distribution with mean a + b and variance

Ex. An assembly consists of 3 linkage components as shown here. The properties X1, X2, X3 are given below with means in cm, variance in cm2.

X1  N (12, 0.02)

X2  N (24, 0.03)

X3  N (18, 0.04)

If the links are produced by different machines and operators, determine P (53.8  Y  54.2)

Gamma Distribution

- very important because it allows us to define the exponential and chi-square

distribution, that are used extensively in applied statistics.

The theoretical basis for the gamma distribution = gamma function:

Definition The gamma function is defined by
=
where > 0

Properties of the Gamma Function

1. If  > 1, then

= 1

3. For any positive integer n,

= (n-1)!

Gamma Distribution

Definition A random variable X has a Gamma Distribution with parameters and  if its density is :
f (x) =
where x > 0
 > 0
 > 0
E(x) = 
Var x = 2

Shape of Gamma Distribution

,  = shape parameters

Exponential Distribution

= Gamma Distribution with = 1

Recall: Gamma Density

f (x) =

Thus, exponential density

f (x) = where x > 0
and  > 0

E (x) = 

Var x = 2

Application of Exponential Distribution

Usually occur in common with Poisson process

If  = mean arrival rate

then t = the time between successive arrivals will have an exponential distribution with  = 1/ 

Since x is the time, we can write f(x) as f(t)

where t > 0 and  > 0

t = the time that an arrival occurs

f (t) = probability of an arrival at time t

Ex. The time T between the arrival of orders at a regional warehouse is recorded for 24 orders: 17 19 25 34 35 35 37 39 40 40 41 41 42 42 44 46 51 52 52 56 71 72 80 93

(a) Estimate and interpret the parameter 

(b) Assume arrivals follow an exponential distribution. Plot  and  on the graph.

Ex. Engineers have collected data from 100 compressors on natural gas pipelines and found that the average life is 5.75 years and that failures follow the exponential distribution.

(a) Compute the probability of failure during the first year after installation, and during the first 3 months.

(b) Compute the probability of failure prior to the average life.

Chi-squared distribution

Use in making inference about variance

Def. Let X be a gamma random variable with  = 2 and for  being a positive integer. X is said to have a chi-squared distribution with  degrees of freedom. We denote this variable by
, x  0

0 , x  0
E (X) = 
Var X = 2

Weibull Distribution

Def. A random variable X is said to have a Weibull distribution with parameters  and  if its density is given by:

where x > 0,  > 0,  > 0
= E (x) =

Shape

- varies depending on the values of  and 

-General shape resembles that of gamma density with the curve becoming more symmetic as the value of  increases.

Uniform Distribution

A uniform variable is a variable whose values have equal probability of occurrence throughout a specified interval.

,  x 
f(x) =
0 , otherwise
E (X) =
V(X) =

Proof E (X) =

V(X) =

Graph of U (x; , )

0 ; x 

F (X) = = ;  x 

1 ; x 

Ex. If a value is randomly chosen from a uniform random variable in the interval [0, 10], determine the probability that the value is between 3/2 and 7/2

Ex. The noise level (N) in a workshop is a uniform random variable between 80-95 db. If the safe limit of noise exposure must not exceed 90 db, find the pdf of the random variable N and determine the probability that the noise level exceeds the safe limit.