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Continuous Random Variables
Continuous random variable
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Definition: A random variable X is continuous if the CDF
FX (x) = P(X ≤ x) is a continuous function of x.
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Example 1: Weight of a new born baby;
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Example 2: Waiting time in a bus stop.
Density function
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Definition: The probability density function fX (x) of a
continuous random variable X is the function that satisfies
Rx
FX (x) = −∞ fX (t)dt.
CDF and density function
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If X has a density fX (x), then FX (x) =
Rx
−∞ fX (u)du.
If
FX (x) is differentiable and fX (x) is continuous at point x0 ,
then
d
FX (x)|x=x0 = FX0 (x0 ) = fX (x0 ).
dx
Example
Suppose X is a continuous random variable whose probability
density function is given by

 C(4x − 2x 2 ), 0 < x < 2;
fX (x) =

0
otherwise;
(a) What is the value of C
(b) Find P(X > 1)?
Example
If X is a continuous random variable whose probability density
function is fX (x) and distribution function is FX (x), Find the
density function of Y = 2X .
Density is not probability
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Note that fX (t) is not probability. Actually, P(X = t) = 0 for
any t if X is a continuous random variable. Because
{X = t} ⊂ {t − < X ≤ t} for any P(X = t) ≤ P(t − < X ≤ t) = FX (t) − FX (t − ).
Hence 0 ≤ P(X = t) ≤ lim→0 [FX (t) − FX (t − )] = 0 by
continuity of FX .
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Any meaningful statement about probability must consider
X lying in some interval. Probability is interpreted as the
area under the density function.
Example
The amount of time in hours that a computer functions before
breaking down is a continuous random variable with probability
density function given by

 λ exp−x/100 ,
x ≥ 0;
fX (x) =

0
otherwise;
What is the probability that
(a) a computer will function between 50 and 150 hours before
breaking down?
(b) it will function for fewer than 100 hours?
Example: Logistic distribution
A random variable X with logistic distribution if FX (x) =
1
.
1+e−x
Then
fX (x) =
dFX (x)
e−x
=
dx
(1 + e−x )2
P(a < X < b) = FX (b) − FX (a)
Z b
Z
=
fX (x)dx −
−∞
a
Z
fX (x)dx =
−∞
If ∆x is small, P(a ≤ x ≤ a + ∆x) ≈ fX (a)∆x.
b
fX (x)dx.
a
Expected value
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Definition: Let X be a continuous random variable with
density f (x). Then the expected value of a random variable
g(X ) is
Z
E(g(X )) =
provided that
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R
g(x)f (x)dx
|g(x)|f (x)dx exists.
Linearity:
E(ag1 (X ) + bg2 (X ) + c) = aE(g1 (X )) + bE(g2 (X )) + c.
Examples
Find E(X ) when the density function of X is

 2x, 0 ≤ x ≤ 1;
fX (x) =
 0 otherwise;
Examples
Find E(eX ) when the density function of X is

 1, 0 ≤ x ≤ 1;
fX (x) =
 0 otherwise;
Examples
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Example 1: If X has exponential(λ), i.e.,
fX (x) =
x
1
exp(− ) x ≥ 0 and λ > 0.
λ
λ
What is the expectation of X ?
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Example 2: If X has Cauchy distribution, the density of X is
fX (x) =
1 1
π 1 + x2
What is the expectation of X ?
− ∞ < x < ∞.
Variance
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For any random variable X , the variance is
Var(X ) = E[(X − E(X ))2 ].
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Example: If X has exponential(λ), what is the variance of
X?