Continuous Probability Distributions Continuous Random Variables & Probability Distributions
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Transcript of Continuous Probability Distributions Continuous Random Variables & Probability Distributions
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Continuous Probability Distributions
Continuous Random Variables &Probability Distributions
Dr. Jerrell T. Stracener, SAE Fellow
Leadership in Engineering
EMIS 7370/5370 STAT 5340 : PROBABILITY AND STATISTICS FOR SCIENTISTS AND ENGINEERS
Systems Engineering ProgramDepartment of Engineering Management, Information and Systems
Stracener_EMIS 7370/STAT 5340_Sum 08_06.05.08
Stracener_EMIS 7370/STAT 5340_Sum 08_06.05.08
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•Definition - A random variable is a mathematical function that associates a number with every possible outcome in the sample space S.
• Definition - If a sample space contains an infinite number of possibilities equal to the number of points on a line segment, it is called a continuous sample space and a random variable defined over this space is called a continuous random variable.
• Notation - Capital letters, usually or , are used to denote random variables. Corresponding lower case letters, x or y, are used to denote particular values of the random variables or .
X Y
YX
Random Variable
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For many continuous random variables or (probabilityfunctions) there exists a function f, defined for allreal numbers x, from which P(A) can for any eventA S, be obtained by integration:
Given a probability function P() which may berepresented in the form of
A
dxxfAP
areaA
dxxfAP
Continuous Random Variable
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in terms of some function f, the function f is calledthe probability density function of the probabilityfunction P or of the random variable , and the probability function P is specified by the probability density function f.
X
Continuous Random Variable
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Probabilities of various events may be obtained from the probability density function as follows:
Let A = {x|a < x < b}
Then
P(A) = P(a < X < b)
A
dxxf
b
a
dxxf
Continuous Random Variable
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Therefore = area under the density function curvebetween x = a and x = b.
f(x)
x
Area = P(a < x <b)
a b0 0
)(AP
Continuous Random Variable
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The function f(x) is a probability density function for the continuous random variable X, defined over the set of real numbers R, if
1. f(x) 0 for all x R.
2.
3. P(a < X < b) =
.1dx)x(f
b
a
.dx)x(f
Probability Density Function
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The cumulative probability distribution function, F(x), of a continuous random variable with density function f(x) is given by
Note:
x
.dt)t(f)xX(P)x(F
xFdx
df(x)
X
Probability Distribution Function
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Probability Density and Distribution Functions
f(x) = Probability Density Function
x
Area = P(x1 < <x2)
F(x) = Probability Distribution Function
x
F(x2)
F(x1)
x2x1
P(x1 < <x2) = F(x2) - F(x1)
1
cumulative area
x2x1
X
X
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• Mean or Expected Value
• Remark
Interpretation of the mean or expected value:The average value of in the long run.
dxxfx XEμ
X
Mean & Standard Deviationof a Continuous Random Variable X
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•Variance of X:
•
•Standard Deviation of :
dx f(x) μ)-(xσXVar 22
22 μXEXVar
22 μxfx dx
XVarσ X
Mean & Standard Deviationof a Continuous Random Variable X
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If a and b are constants and if = E is the meanand 2 = Var is the variance of the randomvariable , respectively, then
and
baμbaXE
XVarabaXVar 2
X)(X
)(X
Rules
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If Y = g(X) is a function of a continuous random variable , then
dxxfxgxgEμY
X
Rules
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If the probability density function of X is
)(xf)1(2 x
0
for 0 < x < 1
elsewhere
then find
(a) and
(b) P(X>0.4)
(c) the value of x* for which P(X<x*)=0.90
Example
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First, plot f(x):
0
0.5
1
1.5
2
0 0.2 0.4 0.6 0.8 1
x
f(x)
Example
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Find the mean and standard deviation of X,
1
0
)()( dxxxfXE
dxxxdxxx 1
0
21
0
][2)1(2
3
1
2
12
322
1
0
32 xx
3
1
3
21
Example Solution
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222 )()( XEXVar
21
0
2
3
1)(
dxxfx
9
1
432
9
1)1(2
1
0
431
0
2
xx
dxxx
9
1
12
2
3
1
4
1
3
12
2
Example Solution
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and the standard deviation is
18
1
12
2
3
1
3
1
4
2
3
1
236.018
1
Example Solution
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(b)
(c)
2
00
2
22 )()()(
xx
dxxdxxfxXPxFxx
)4.0X(P1)4.0X(P
36.064.01
4.04.0*21 2
1.32or 0.68 x*
9.0*)(*)(2*)(*)( 2
therefore
xxxPxXP
Example Solution
for 0<x<1
Since 1.32>1, so 0.68x*
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Uniform Distribution
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Probability Density Function
elsewhere , 0
0afor b, x afor , 1
)( abxf
a b0
f(x)
x
1/(b-a)
Uniform Distribution
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Probability Distribution Function
b x afor ab
ax )()(
xXPxF
a b0
F(x)
x
1
a for x 0
b for x 1
Uniform Distribution
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• Mean
= (a+b)/2
• Standard Deviation
12
ab
Uniform Distribution
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Example – Uniform Distribution
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Example Solution – Uniform Distribution