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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(1p)>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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