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K. Desch – Statistical methods of data analysis SS10 2. Probability 2.3 Joint p.d.f.´s of several...

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Transcript of K. Desch – Statistical methods of data analysis SS10 2. Probability 2.3 Joint p.d.f.´s of several...
K. Desch – Statistical methods of data analysis SS10
2. Probability 2.3 Joint p.d.f.´s of several random variables
Examples: Experiment yields several simultaneous measurements(e.g. temperature and pressure)
Joint p.d.f. (here only for 2 variables):
f(x,y) dx dy = probability, that x[x,x+dx] and y[y,y+dy]
Normalization:
Individual probability distribution (“marginal p.d.f.”) for x and y:
yields probability density for x (or y) independent of y (or x)
x and y are statistically independent if
Sf(x,y) dxdy 1
xf (x) f(x,y) dy
yf (y) f(x,y) dx
x yf(x,y) f (x) f (y)
f(x  y) f(x) for any y (and vice versa)
K. Desch – Statistical methods of data analysis SS10
2. Probability 2.3 Joint p.d.f.´s of several random variables
conditional p.d.f.´s:
h(yx)dxdy is the probability for event to lie in the interval [y,y+dy]when the event is known to lie in the interval [x,x+dx].
x
f(x,y)h(y  x)
f (x)
y
f(x,y)g(x  y)
f (y)
K. Desch – Statistical methods of data analysis SS10
2. Probability 2.3 Joint p.d.f.´s of several random variables
Example: measurement of the length of a bar and its temperature
x = deviation from 800 mmy = temperature in 0C
a) 2dimentional histogram (“scatterplot”)
b) Marginal distribution of y (“yprojection”)
c) Marginal distribution of x (“xprojection”)
d) 2 conditional distributions of x (s. edges in (a))
Width in d) smaller than in c)
x and y are “correlated”
K. Desch – Statistical methods of data analysis SS10
2. Probability 2.3 Joint p.d.f.´s of several random variables
Expectation value (analog to 1dim. case)
n1E[a(x)] a(x) f(x)dx ...dx
Variance (analog to 1dim. case)
2 2a a n1
V[a(x)] (a(x) ) f(x)dx ...dx
Covariance
for 2 variables x, y with joint p.d.f. f(x,y):
important when more than one variable: measure for the correlation of the variables:
xy x ycov[x,y] V : E[(x )(y )]
x yE[xy]
x y... xy f(x,y)dxdy
if x, y are stat. independent (f(x,y) = fx(x)fy(y)) then cov[x,y] = 0(but not vice versa!!)
K. Desch – Statistical methods of data analysis SS10
2. Probability 2.3 Joint p.d.f.´s of several random variables
Positive correlation: positive (negative) deviation of xfrom its x increases the probability, that y has a positive (negative) deiationof its y
For the sum of random numbers x+y holds: V[x+y] = V[x] + V[y] + 2 cov[x,y](proof: linearity of E[])
For n random variables xi i=1,n: is the covariance matrix (symmetric matrix)
diagonal elements:
For uncorrelated variables: covariance matrix is diagonal
For all elements:
i
2i i i xcov[x ,x ] V[x ]
i ji j x xcov[x ,x ] V
i ji j x xcov[x ,x ]
Normalized quantity: is the correlation coefficient i j
i j
i jx x
x x
cov[x ,x ]:
K. Desch – Statistical methods of data analysis SS10
2. Probability 2.3 Joint p.d.f.´s of several random variables
examples for correlation coefficients
(Axis units play no role !)
K. Desch – Statistical methods of data analysis SS10
2. Probability 2.3 Joint p.d.f.´s of several random variables
one more example:
[Barlow]
K. Desch – Statistical methods of data analysis SS10
2. Probability 2.3 Joint p.d.f.´s of several random variables
another example:
K. Desch – Statistical methods of data analysis SS10
2. Probability 2.4 Transformation of variables
Measured quantity: x (distributed according to pdf f(x))
Derived quanitity: y = a(x) What is the p.d.f. of y, g(y) ?
Define g(y) by requiring the same probability for
y
x
[y,y+dy]
[x,x+dx] =: dS
a(x)
dS
g(y)dy f(x)dx
dxg(y) f(x(y))
dy
K. Desch – Statistical methods of data analysis SS10
2. Probability 2.4 Transformation of variables
More tedious when x y not a 11 relation, e.g.2xy(x)
y
[y,y+dy]
;y2
1
dy
dx )yf()yf(
y2
1g(y)
two branches x>0 and x<0
for g(y) sum up the probabilitiesfor x>0 and x<0
K. Desch – Statistical methods of data analysis SS10
2. Probability 2.Transformation of variables
Functions of more variables :
transformation through the Jacobian matrix:
g(y) f(x(y))det J
1 1
1 n
n n
1 n
x x
y y
J
x x
y y
y a(x)
K. Desch – Statistical methods of data analysis SS10
2. Probability 2.4 Transformation of variables
Example: Gaussian momentum distributionMomentum in x and y:
polar coordinates
x = r cos φ
y = r sin φ
r2 := x2 + y2
2xef(x) 2yef(y)
22 yxef(x)
cos rsin
sin rcosy
r
y
x
r
x
J
det J = r → g (r,φ) = f ( x (r,φ), y (r,φ) ) • det J = re2r
In 3dimenions → Maxwell distribution
K. Desch – Statistical methods of data analysis SS10
2. Probability 2.5 Error propagation
Often, one is not interested in complete transformation of p.d.f. but onlyin the transformation of its variance (=squared error) measured error of x derived error of y
When σx is small relative to curvature of y(x) :
→ linear approach
E y(x) y(μ)
n
i ii 1 i x
yy(x) y( ) (x )
x
What about the variance?
K. Desch – Statistical methods of data analysis SS10
2. Probability 2.5 Error propagation
Variance : 2) )μy()xy( (E )xy(V
2n
1iiiμx
i
)μ(xx
yE
n
1jjjμx
j
n
1iiiμx
i
)μ(xx
y)μ(x
x
yE
n n2y ij
i 1 j 1 i j
y yV[y] V
x x→
K. Desch – Statistical methods of data analysis SS10
2. Probability 2.5 Error propagation
For more variables yi:
n nk l
k l i ji 1 j 1 i j
y ycov[y ,y ] cov[x ,x ]
x x
→ general formula for error propagation (in linear approximation)
Special cases:
a) uncorrelated xj :
and
x
y )xy(V
n
1iμx
2x
2
ii
σx
y
x
y y,ycov
n
1iμx
2x
i
l
i
klk i
even if xi are uncorrelated, the yi are in general correlated
K. Desch – Statistical methods of data analysis SS10
2. Probability 2.5 Error propagation
b) Sum y = x1 + x2 → 2x
2x
2y 21
σσσ
errors added in quadratures
c) Product y = x1x2 → 2x
2x
2x
2x
2y
2y
2y 2
2
1
1
μ
σ
μ
σ
μ
σ
μ
v[y]
relative errors added in quadratures
x1 and x2 are uncorrelated !
K. Desch – Statistical methods of data analysis SS10
2. Probability 2.5 Convolution
Convolution :
Typical case when a probability distribution consists of two random variables x, y like a sum w = x + y. w is also a random variable
Example: x: BreitWigner Resonance
y: Exp. Resolution (Gauss)
What is the p.d.f. for w when fx(x) and fy(y) are known
w x yf (w) f (x)f (y)δ(w x y)dxdy
x)dx(w(x)ff yx
y)dy(w(y)ff xy
yx ff
xy
3. Distributions
Important probability distributions
 Binominal distribution
 Poisson distribution
 Gaussian distribution
 Cauchy (BreitWigner) distribution
 Chisquared distribution
 Landau distribution
 Uniform distribution
Central limit theorem
3. Distributions 3.1 Binomial distribution
Binomial distribution appears when one has exactly two possible trial outcomes (successfailure, headtail, evenodd, …)
event “success”: event “failure”:
Probability:
A A
p P(A) q (1 p) P(A)
Example: (ideal) coins
Probability for “head” (A) = p = 0.5, q=0.5
Probability for n=4 trials to get ktime “head” (A) ?
k=0: P = (1p)4 = 1/16k=1: P = (p (1p)3) times number of combinations (HTTT, THTT, TTHT, TTTH) = 4*1/16 = ¼k=2: P = (p2 (1p)2) times (HHTT, HTTH, TTHH, HTHT, THTH, THHT) = 6*1/16 = 3/8 k=3: P = (p3 (1p)) times (HHHT, HHTH, HTHH, THHH) = 4*1/15 = ¼k=4: P = p4 = 1/16
P(0)+P(1)+P(2)+P(3)+P(4) = 1/16+1/4+3/8+1/4+1/16 = 1 ok
3. Distributions 3.1 Binomial distribution
Number of permutations for k successes by n trials:
Binominal coefficient:
Binomial distribution:
 Discrete probability distribution
 Random variable: k
 Depends on 2 parameters: n (number of attempts) and p (probability of suc.)
 Sequence of appearance of k successes play no role
 n trials must be independent
n n!
k k!(n k)!
k n k n
f(k;n,p) p (1 p)k
3. Distributions 3.1 Binomial distribution (properties)
Normalisation:
Expectation value(mean value):
Proof:
n nk n k n
k 0 k 0
nf(k;n,p) p (1 p) (p (1 p)) 1
k
n nk n k k 1 n k
k 0 k 1
nk 1 n k
k 1
nk n k
k 0
n! (n 1)!kp (1 p) np kp (1 p)
k!(n k)! k!(n k)!
(n 1)!np p (1 p)
(k 1)!(n k)!
n !np p (1 p) np (mit n n 1,k k 1)
k !(n k )!
npp)n,kf(k;kE[k]k
3. Distributions 3.1 Binomial distribution (properties)
Variance:
Proof:
nk n k
k 0
n2 k 2 n k
k 2
n2 k n k 2
k 0
n!k(k 1)p (1 p)
k!(n k)!
(n 2)!p n(n 1) p (1 p)
(k 2)!(n k)!
n !p n(n 1) p (1 p) p n(n 1) (mit n n 2,k k 2)
k !(n k )!
k(k 1)
However:
2 2 2 2 2 2
2 22 2 2 2 2
V[k] E[k ] E[k] E[k ] E[k] E[k] E[k] E[k k] E[k] E[k ]
k k k k p n(n 1) np n p np(k 1k p)
p)np(1p)n,f(k;μ)(kσV[k]k
22
3. Distributions 3.1 Binomial distribution
3. Distributions 3.1 Binomial distribution
HERAB experiment muon spectrometer
12 chambers; efficiency of one chamber is ε = 95%Trigger condition: 11 out of 12 chambers hit
εTOTAL = P(11; 12,0.95) + P(12; 12,0.95) = 88.2 %
When chambers reach only ε = 90% then εTOTAL = 65.9%
When one chambers fails: εTOTAL = P(11, 0.95, 12) = 56.9 %
Random coincidences (noise): εBG = 10% 20%  twice more noise εTOTAL_BG = 1•109 2•107 200x more background
x x x x
xxxxxxx
μ
3. Distributions 3.1 Binomial distribution
Example: number of error bars in 1interval (p=0.68)