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Probability theory Random variables. In an experiment a number is often attached to each outcome. S. s. X(s). R. Definition: A random variable X is a function defined on S, which takes values on the real axis. X: S R. Sample space. Real numbers. - PowerPoint PPT Presentation
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lecture 21
Probability theoryRandom variables
Definition: A random variable X is a function defined on S, which takes values on the real axis
In an experiment a number is often attached to each outcome.
X: S R
Real numbersSample space
R
S
X(s)s
lecture 22
Probability theoryRandom variables
Example:
Random variable Type
Number of eyes when rolling a dice discreteThe sum of eyes when rolling two dice discreteNumber of children in a family discreteAge of first-time mother discreteTime of running 5 km continuousAmount of sugar in a coke continuous Height of males continuous
counting
measure
Discrete: can take a finite number of values or an infinite but countable number of values.
Continuous: takes values from the set of real numbers.
lecture 23
Discrete random variableProbability function
Definition: Let X : S R be a discrete random variable.
The function f(x) is a probabilty function for X, if
1. f(x) 0 for all x 2.
3. P(X = x) = f(x),
where P(X=x) is the probability for the outcomes sS : X(s) = x.
x
1)x(f
lecture 24
Discrete random variableProbability function
Example: Flip three coins X : # heads X : S {0,1,2,3}
Probability fuinction
f(0) = P(X=0) = 1/8
f(1) = P(X=1) = 3/8
f(2) = P(X=2) = 3/8
f(3) = P(X=3) = 1/8
Value of X X=0 X=1 X=2 X=3
Outcome TTT
HTT, TTH, THTHHT, HTH, THHHHH
Notice! The definition of a probability function is fulfilled:
1. f(x) 02. f(x) = 13. P(X=x) = f(x)
lecture 25
Discrete random variableCumulative distribution function
Definition:
Let X : S R be a discrete random variable with probability function f(x).
The cumulative distribution function for X, F(x), is defined by
F(x) = P(X x) = for - < x <
xt
)t(f
lecture 26
Discrete random variableCumulative distribution function
Example: Flip three coins X : # heads X : S {0,1,2,3}
Cumulative dist. Func.
F(0) = P(X < 0) = 1/8F(1) = P(X < 1) = 4/8F(2) = P(X < 2) = 7/8F(3) = P(X < 3) = 1
Value of X
X=0 X=1 X=2 X=3
Outcome
TTTHTT, TTH, THTHHT, HTH, THHHHH
Probability function
f(0) = P(X=0) = 1/8f(1) = P(X=1) = 3/8f(2) = P(X=2) = 3/8f(3) = P(X=3) = 1/8
Cumulative distribution function:Probability function:
x0 1 2 3
0.10.20.30.4
x0 1 2 3
0.20.40.60.8
f(x) F(x) 1.0
lecture 27
Continuous random variable
A continuous random variable X has probability 0 for all outcomes!!
Mathematically: P(X = x) = f(x) = 0 for all x
Hence, we cannot represent the probability function f(x) by a table or bar chart as in the case of discrtete random variabes.
Instead we use a continuous function – a density function.
lecture 28
Definition: Let X: S R be a continuous random variables.
A probability density function f(x) for X is defined by:
1. f(x) 0 for all x
2.
3. P( a < X < b ) = f(x) dx
Note!! Continuity: P(a < X < b) = P(a < X < b) = P(a < X < b) = P(a < X < b)
Continuous random variableDensity function
b
a
1dx)x(f
lecture 29
Probability of a service life longer than 3 years:
Example: X: service life of car battery in years (contiuous)
Density function:
Continuous random variableDensity function
0
0,1
0,2
0,3
0,4
0,5
1 3,5 63
otherwise
xforx
xforx
xf
0
65.316.096.0
5.3116.016.0
)(
68.0
)16.096.0()16.016.0(
)()3(
6
5.3
5.3
3
3
dxxdxx
dxxfXP
lecture 210
Probability of a service life longer than 3 years:
Alternativ måde:
Continuous random variableDensity function
othwerwise
xforx
xforx
xf
0
65.316.096.0
5.3116.016.0
)(
0
0,1
0,2
0,3
0,4
0,5
1 3,5 63
68.032.01
)16.016.0(1
)(1
)3(1)3(
3
1
3
dxx
dxxf
XPXP
lecture 211
Continuous random variableDensity function
Definition: Let X : S R be a contiguous random variable with density function f(x).
The cumulative distribution function for X, F(x), er defined by
F(x) = P(X x) = for - < x < Note: F´(x) = f(x)
32.0
dxx16.016.0
dx)x(f
)3X(P)3(F
3
1
3
0
0,1
0,2
0,3
0,4
0,5
1 3,5 63
x
dt)t(f
lecture 212
Continuous random variableDensity function
0
0,1
0,2
0,3
0,4
0,5
1 3,5 6
From the definition of the cumulative distribution funct. we get:
P( a < X < b ) = P( a < X < b ) = P( a < X < b ) = P( a < X < b ) = P(X < b ) – P(X < a )
= F(b) – F(a)
a b
lecture 213
Continuous random variableDensity function
Contiuous random variable • The sample space contains infinitely many outcomes
• Density function f(x) is a Continuous function
• Calculation of probabilities
P( a < X < b ) = f(t) dt
Discrete random variable
• Sample space is finite or has countable many outcomes
• Probability function f(x)Is often given by table
• Calculation of probabilitiesP( a < X < b) = f(t)
a<t<bb
a
lecture 214
Joint distribution Joint probability function
Definition: Let X and Y be two discrete random variables. The joint probability function f(x,y) for X and YIs defined by
1. f(x,y) 0 for all x og y
2.
3. P(X = x ,Y = y) = f(x,y) (the probability that both X = x and Y=y)
For a set A in the xy plane:
x y
1)y,x(f
)y,x( A
)y,x(f)A)Y,X((P
lecture 215
Joint distribution Marginal probability function
Definition: Let X and Y be two discrete random variables with joint probability function f(x,y).The marginal probability function for X is given by
g(x) = f(x,y) for all x
The marginal probability function for Y is given by
h(y) = f(x,y) for all y
y
x
lecture 216
Joint distribution Marginal probability function
Example 3.14 (modified):The joint probability function f(x,y) for X and Y is given by
2
3/28
0
0
0
1
2
1
9/28
3/14
0
x 0
3/28
3/14
1/28
y• h(1) = P(Y =1) = 3/14+3/14+0 = 3/7
• g(2) = P(X= 2) = 3/28+0+0 = 3/28
• P(X+Y < 2) = 3/28+9/28+3/14 = 18/28 = 9/14
lecture 217
Joint distributionJoint density function
Definition: Let X og Y be two continuous random variables. The joint density function f(x,y) for X and Y is defined by
1. f(x,y) 0 for all x
2.
3. P(a < X< b, c<Y< d) =
For a region A in the xy-plane: P[(X,Y)A] = f(x,y) dxdyA
1dydx)y,x(f
d
c
b
adydx)y,x(f
lecture 218
Definition: Let X and Y be two continuous random variables with joint density function f(x,y).The marginal density function for X is given by
The marginal density function for Y is given by
Joint distributionMarginal density function
xallfordyyxfxg
),()(
yallfordxyxfyh
),()(
lecture 219
Joint distributionMarginal density function
Example 3.15 + 3.17 (modified):Joint density f(x,y) for X and Y:
Marginal density function for X:
otherwise0
1y01,x03y)(2x5
2y)f(x,
53
x541
051
52
52
2
1
0
y3xy2
dy)y3x2(
dy)y,x(f)x(g
0
0.5
1
00.2
0.40.6
0.810
0.5
1
1.5
2
lecture 220
Joint distributionConditional density and probability functions
Definition: Let X and Y be random variables (continuous or discrete) with joint density/probability function f(x,y). Then the
Conditional denisty/probability function for Y given X=x is
f(y|x) = f(x,y) / g(x) g(x) 0
where g(x) is the marginal density/probability function for X, and
the conditional density/probability function for X given Y=y is f(x|
y) = f(x,y) / h(y) h(y) 0
where h(y) is the marginal density/probability function for Y.
lecture 221
Joint distributionConditional probability function
Examples 3.16 + 3.18 (modified):Joint probability functionf(x,y) for X and Y is given by:
2
3/28
0
0
0
1
2
1
9/28
3/14
0
x0
3/28
3/14
1/28
y
• marginal pf.
• P(Y=1 | X=1 ) = f(1|1) = f(1,1) / g(1) = (3/14) / (15/28) = 6/15
2xfor
1xfor
0xfor
)x(g
28328152810
lecture 222
Joint distributionIndependence
Definition: Two random variables X and Y (continuous or discrete) with joint density/probability functions f(x,y) and marginal density/probability functions g(x) and h(y), respectively, are said to be independent if and only if
f(x,y) = g(x) h(y) for all x,y
or if f(x|y) = g(x) (x indep. of y) or f(y|x)=h(y) (y indep. af x)