Example 7: Solving a simultaneous integration
A continuous random variable X, has a probability distribution function given
by:
⎧ +
a
bx
+
x
2
for
1
≤
x
≤
4
f x
( )
= ⎨
⎪
9
⎪⎩
0
elsewhere
(
Given that
P x
≤
2
)
=
5
, find the values of a and b.
27
4
From the fact that it is a probability distribution function we know that
f x dx
=
1
. We can use this to get
( )
∫
1
the first expression in terms of a and b.
As can be seen above, the initial integration gives an equation involving a fraction. This can be simplified by
multiplying both sides by 6, highlighting the entire equation first. Notice that when the final simplification is
equal to zero, the calculator does not bother to include the ‘
=0
’. All expressions are assumed to be equal to
zero unless otherwise specified.
2
The second probability tells us that
f x dx
=
and this
( )
5
∫
1
27
gives the second of the pair of simultaneous equations in
exactly the same fashion.
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