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| PAGE: 598 | SET: Exercises | PROBLEM: 8 |
(a)
If 
is a probability density function then it must satisfy 
.
Observe the graph:
The graph of 
is a triangular region.
Find 
.
Calculate the area under the graph of with 
from 
to 
.
The triangle height is 
units and base is 
units.
Area of the triangle 
.
.
and 
for all 
.
Therefore, the function 
whose graph is shown is a probability density function.
(b)
(i)
Find 
.
.
.
From the graph, 
.
Find 
.
Calculate the area under the graph of with 
from 
to 
.
The triangle height is 
units and base is 
units.
.
.
(b)
(ii)
Find 
.
.
Calculate the area under the graph of with from to 
.
From part (b)(i) .
.
Estimate the area under the graph of with from 
to .
The triangle height is 
units and base is 
units.
.
.
.
(c)
The mean of any probability density function 
is defined to be 
.
.
Observe the graph:
Find 
.
if 
.
.
Find 
.
if 
.
.
The mean is 
.
.
(a), whose graph is shown is a probability density function.
(b) ; .
(c) .
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