Posted: February 1st, 2016

A random variable X can be described by its density function fx(t) or its distribution function Fx(t). They are related as follows.

A random variable X can be described by its density function fx(t) or its distribution function Fx(t). They are related as follows.
∫ ∞−
=
=
t
xx
xx
(u)duf(t)F
(t)F
dt d
(t)f
The first two moments of a random variable X are
∫
∫
∞
∞−
∞
∞−
⋅=
⋅=
(t)dtft)E(X
(t)dtftE(X)
x
x
22
The convolution g(t) is defined as
∫ ∞ ∞− −⋅= u)du f (t(u)fg(t) xx
Later, these formulas will be given physical interpretations; but for now, take the viewpoint that the following exercise will get the mathematical details out of the way so that later we can concentrate on their meaning rather than their calculation.
Fill in the following table. Show all work on separate pages. Draw the graphs of fx(t), Fx(t) and g(t) on the graph paper provided.
Take a = 0.7 and b = 1.5. In the second case, write the answers in terms of the parameter
λ , but use 2 =λ when
drawing the graphs and evaluating ∫b a
x(t)dtf and ∫b a
g(t)dt .