Math 419 – Spring 2016 Transformations and Limits Problem Set 11

```Math 419 – Spring 2016
Transformations and Limits
Problem Set 11
Problems
1. Losses for an insurance company follow a continuous distribution with probability density function
f(x) = 375 / (x + 5)4 , for x &gt; 0.
What is the probability density function of Y = 1.3X + 0.4?
1
Math 419 – Spring 2016
Transformations and Limits
Problem Set 11
2. The accumulated value of an investment of 10,000 is modeled as a random variable Y = 10000e2X ,
where X is a continuous random variable with probability density function fX(x) = Ce– x, for
0 &lt; x &lt; 1, and zero otherwise, and where C is a positive constant. Find fY(y), the pdf of Y, and the
region where it is positive.
2
Math 419 – Spring 2016
Transformations and Limits
Problem Set 11
3. The time T that a manufacturing system is out of operation has a cumulative distribution function
F(t) = 1 – (2/t)2 , for t &gt; 2 , and zero otherwise. The resulting cost to the company Y = T2.
Determine the density function of Y, for y &gt; 4.
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Math 419 – Spring 2016
Transformations and Limits
4. Define the cumulative distribution function of X as follows:
F(x) =
0, if x &lt; 0.5
x2, if 0.5 &pound; x &lt; 1
1, if x &sup3; 1
Let Y = 1/ (3X). Find E(Y).
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Problem Set 11
```