Probability Density Function (pdf)
Exercise 2-3.1
The pdf for a random distribution is given by the function:
( )
( )
probability distribution function F(x)
1
F(x)
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.2
1.4
x
probabilty density function
5
fx(x)
4
3
2
1
0
0
0.2
0.4
0.6
0.8
1
x
Initially we want to solve for the unknown constant (k),
which will be done through integration of the pdf.
∫
( )
Since the step function u(x) is zero for any number less
Than 0 and 1 elsewhere the integration of the function becomes:
∫
We can use integration to solve for the probability of a range of
values. Initial we are going to solve for the probability of X
falling between 0 and 0.5. Inspecting the probability distribution
graph we can see by the time we reach 0.5 a fair measure of the
probability falls inside this range.
(
)
∫
(
)
∫
( )
Next we want to solve for the probability X is greater than 1.
By inspection of the graph of F(x) we can see not much there isn’t much
probability left to accumulate. The overall change of X > 1 should not
be very high. First we must find the probability of X less than or equal
to 1.
(
(
)
∫
(
)
∫
( )
)
Now we have the probability of less than or equal to 1.
By subtracting from 1 we will solve for the remaining
probability.
(
)
(
)
Exercise 2-3.2
We can reshape the pdf if we know the relationship between
X and Y. For this problem the relationship is:
The pdf of X and Y can be found through the relationship:
( )
( )
( )
( )
We want to take the derivative of the relationship between
X and Y with respect to Y. The equation will need to be rearranged.
( )
( )
Replace any x with the new relationship:
( )
(
(
)
)
The step function is 1 or 0 depending on the difference between
Y and a constant, in this case -3. The 5 in the denominator does not
have any meaning in the step function.
(
)
( )
(
)
probability distribution function F(y)
1
F(y)
0.8
0.6
0.4
0.2
0
0
2
4
6
8
10
8
10
x
probabilty density function f(y)
1.2
1
fx(y)
0.8
0.6
0.4
0.2
0
0
2
4
6
x