Probability Distribution of a
Random Variable
The [cummulative] distribution function Fx(t) fully characterizes the
probability distribution of a random variable x. It is defined as:
Fx (t ) = P( ~
x ≤ t)
Obviously Fx(t) indicates the probability whit which a random
variable x will take values smaller or equal to an arbitrary value t.
Since a probability measure can neither be lower than 0 nor it can
be higher than 1, Fx(t) must satisfy the following condition:
0 ≤ Fx (t ) ≤ 1
Financial Services: Applications
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Probability Distribution of a
Random Variable
The probability function (density) of a random variable x, denoted
by fx(t), is defined as the first derivative of Fx(t). Therefore the area
under fx(t) between -∞ and any particular value s exactly
corresponds to the actual value of function Fx(t=s).
s
∂Fx (t )
f x (t ) =
∂t
Fx ( s ) =
∫f
x
(t )dt
−∞
As s → ∞ the probability P(x ≤ s) converges to 1, and hence, the
total area under fx(t) must therefore also be equal to 1, i.e.
∞
∫f
x
(t )dt = 1
−∞
Financial Services: Applications
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Probability Distribution of a
Random Variable
Since fx(t) is the first derivative of Fx(t) or equivalently Fx(t) is the
indefinite integral of fx(t), it automatically follows that:
b
F (b) − F (a ) = ∫ f (t )dt
a
Verbally this means the following: If fx(t) is the probability density
of a random variable x, then the area under fx(t) between any two
values a and b [= the definite integral of fx(t) between the values a
and b] corresponds exactly to the probability that x will take some
value between a and b!
Financial Services: Applications
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Probability Distribution of a
Random Variable
Since fx(t) exactly declares the probability of a random variable x
to take any [set of] value[s], it can well be applied to derive the
expected value of an arbitrary function of x. In general, one can
say:
∞
Ε[g ( ~
x )] = g (t ) f (t )dt
∫
x
−∞
Thus, it must also hold for the following special cases:
∞
Ε(~
x ) = ∫ tf x (t )dt
{
∞
−∞
} ∫ [t − Ε(~x )]
2
Var (~
x ) = Ε [~
x − Ε(~
x )] =
2
−∞
∞
2
f x (t )dt = ∫ t 2 f x (t )dt − [Ε(~
x )]
−∞
Financial Services: Applications
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