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February 2, 2018 | Author: Anonymous | Category: Math, Statistics And Probability, Normal Distribution
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Continuous distributions Five important continuous distributions: 1. uniform distribution (contiuous) 2. Normal distribution 3. c2 –distribution

[“ki-square”]

4. t-distribution 5. F-distribution 1

lecture 5

A reminder Definition: Let X: S  R be a continuous random variable. A density function for X, f (x), is defined by: 1. f (x)  0 for all x 2.



f ( x ) dx  1



3. P ( a  X  b ) 

b

 a

2

P(a x=90:1:150; y=normpdf(x,120,10); plot(x,y) m s

200

lecture 5

Normal distribution Mean & variance Theorem: If X ~ N(m,s2) then • mean of X:

E(X )  m

• variance of X:

Var ( X )  s

2

Density function: 0.4 0.3

N(0,1) N(1,1) N(0,2)

0.2 0.1 0.0

7

-4

-2

0 x

2

4

lecture 5

Normal distribution Standard normal distribution Standard normal distribution: Z ~ N(0,1) Density function: Distribution function: 

1

f (z) 

2

e

1

z

z

2

F ( z)  P (Z  z) 

2





1 2



e

1

z

2

0.4

(see Table A.3)

0.3 0.2

Notice!! Due to symmetry P(Z  z) = 1 P(Z  z)

0.1 0.0 -3

-2

-1

0

1

2

3

P (1  Z  2)  8

2



1

1 2



e

1 2

z

2

dz lecture 5

2

dx

Normal distribution Standard normal distribution Standard normal distribution, N(0,1), is the only normal distribution for which the distribution function is tabulated. We typically have X~N(m,s2) where m  0 and s2  1. Example:

X ~ N(1,4)

0.4 0.3

What is P( X > 3 ) ?

0.2 0.1 0.0 -2

9

0

2

4

lecture 5

Normal distribution Standard normal distribution Theorem: Standardise X m If X ~ N(m,s2) then ~ N 0 ,1  s

Example cont.: X ~ N(1,4). What is P(X >3) ? P ( X > 3)  P (

X 1

>

2

3 1

)  P ( Z > 1)  1  P ( Z  1)  0 . 1587

2

0.4

Z ~ N(0,1) X ~ N(1,4)

0.3

0.4

0.2

Equal areas = 0.1587

0.2

0.1

0.1

0.0

0.0

-2

10

0

2

Z ~ N(0,1)

0.3

4

-2

0

2

lecture 5

4

Normal distribution Standard normal distribution 0.4 0.3 0.2 0.1 0.0 -2

P ( Z  1)  0 . 8413 11



0

2

4

P ( X > 3 )  1  P ( Z  1)  1  0 .8413  0 .1587 lecture 5

Normal distribution Standard normal distribution Example cont.: X ~ N(1,4). What is P(X >3) ? P ( X > 3 )  1  P ( X  3 )  0 . 1587

Solution in Matlab: >> 1 – normcdf(3,1,2) ans = 0.1587 Solution in R: > 1 - pnorm(3,1,2) [1] 0.1586553

12

Cumulative distribution function normcdf(x,m,s)

Cumulative distribution function pnorm(x,m,s)

lecture 5

Normal distribution Example Problem: The lifetime of a light bulb is normal distributed with mean 800 hours and standard deviation 40 hours:

0.010 0.008 0.006 0.004 0.002 0.000 650

700

750

800

850

900

1. Find the probability that the lifetime of a bulb is between 750-850 hours. 2. Find the number of hours b, such that the probability of a bulb having a lifetime longer than b is 90%. 3. Find a time period symmetric around the mean so that the probability of a lifetime in this interval has probability 95% 13

lecture 5

950

Normal distribution Example Solution in Matlab: 1. P(750 < X < 850) = ? >> normcdf(850,800,40)-normcdf(750,800,40) 2. ? 3. ... Solution in R: 1. P(750 < X < 850) = ? > pnorm(850,800,40)-pnorm(750,800,40) 2. P(X > b) = 0.90  P(X  b) = 0.10 > qnorm(0.1,800,40) 3. ... 14

lecture 5

Normal distribution Relation to the binomial distribution If X is binomial distributed with parameters n and p, then np (1  p )

Rule of thumb: If np>5 and n(1p)>5, then the approximation is good 15

0 .0 0

0 .0 5

is approximately normal distributed.

0 .1 5

X  np

0 .1 0

Z 

0

5

10

15

lecture 5

20

Normal distribution Linear combinations Theorem: linear combinations If X1, X2,..., Xn are independent random variables, where X

~ N ( μ i ,σ i ), for i  1 ,2 , ,n , 2

i

and a1,a2,...,an are constant, then the linear combination Y  a 1 X 1  a 2 X 2    a n X n ~ N ( m Y , s Y ), 2

where m Y  a1 m 1  a 2 m 2    a n m n s Y  a1 s 1  a 2 s 2    a n s n 2

16

2

2

2

2

2

2

lecture 5

The c2 distribution Definition Definition: (alternative to Walpole, Myers, Myers & Ye) If Z1, Z2,..., Zn are independent random variables, where

for i =1,2,…,n,

Zi ~ N(0,1),

then the distribution of n

2

2

2

Y  Z1  Z 2    Z n 



Zi

2

i 1

2

is the c -distribution with n degrees of freedom. Notation: 17

Y ~ c (n) 2

Critical values: Table A.5 lecture 5

The c2 distribution Definition Definition: A continuous random variable X follows a c2distribution with n degrees of freedom if it has density function 1 n 2 1  x 2 f ( x) 

2

n 2

 (n 2)

e

Y ~ c2(1)

0.5 0.4

Y ~ c2 (3)

0.3 0.2

Y ~ c2 (5)

0.1 0.0 0

18

x

2

4

6

8

10

for x > 0

Antag Y ~ c2(n)

E(Y) Var(Y)

=n = 2n

E(Y/n) = 1 Var(Y/n) = 2/n

lecture 5

t-distribution Definition Definition: Let Z ~ N(0,1) and V ~ c2(n) be two independent random variables. Then the distribution of T 

Z V n

is called the t-distribution with n degrees of freedom. Notation: T ~ t(n) 19

Critical values: Table A.4 lecture 5

t-distribution Compared to standard normal 0.4

X ~ N(0,1)

0.3

T ~ T(3)

0.2

T ~ T(1)

0.1 0.0 -3

-2

-1

0

1

2

3

•The t-distribution is symmetric around 0 •The t-distribution is more flat than the standard normal •The more degrees of freedom the more the tdistribution looks like a standard normal 20

lecture 5

F-distribution Definition Definition: Let U ~ c2(n1) and V ~ c2(n2) be two independent random variables. Then the distribution of U F 

n1 V n2

is called the F-distribution with n1 and n2 degrees of freedom. Notation: F ~ F(n1 , n2) 21

Critical values: Table A.6 lecture 5

F-distribution Example

F ~ F(20,50) 1.0

F ~ F(10,30)

0.8 0.6

F ~ F(6,10)

0.4 0.2 0.0 0

22

1

2

3

4

lecture 5

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