Exercise 1. Find the probability density function for the sum of four inde- pendent uniform random variables. This exercise takes a little effort. Nonetheless, one can cut down on the drudge work by being a little clever.
Exercise 1. Find the probability density function for the sum of four inde- pendent uniform random variables. This exercise takes a little effort. Nonetheless, one can cut down on the drudge work by being a little clever.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Exercise 1
Transcribed Image Text:Example
We will find the density for U1+ U2 + U3, for U1, U2, U3 ~ U(0, 1).
To use what we have already done, let's take the density f to be the triangular
density for U1 + U2, given by Equation (2), and the density g = x(0,1)(4) for
random variable U3.
Now one notes that
9(z – x) = X(z-1,2)(x)
(12)
So the convolution integral is
(f * 9)(z) = | f(x)X(z-1,2)(T)dx
(13)
![Exercise 1. Find the probability density function for the sum of four inde-
pendent uniform random variables.
This exercise takes a little effort. Nonetheless, one can cut down on the
drudge work by being a little clever.
• The density is four third degree polynomials spliced together.
• The third degree polynomial on [0, 1] is easy to find, and you can prob-
ably even guess what it is looking at the example leading up to this
exercise.
• The density, as well as its first and second derivatives are continuous
at z = 1; note also that the derivative at z = 2 is zero. On [1,2], the
density has the form h(z) = az° +bz² +cz + d; and the four conditions
noted allow one to set up four linear equations in the four coefficients,
a, b, c, d.
• The density is clearly symmetric about the mean, so once ong finds the
density on [0, 2], finding the density on [2, 4] is immediate. One just has
to think out what the algebraic transformation implied by symmetry
must be. Again, looking at the previous results should help.
A graph of the density is presented in Figure 3. Superimposed on the graph
is a graph of the normal density with the same mean and variance: µ =
+++ = 2 and o? = + ++++
%3D
12](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F718b1378-40e4-4c32-83bc-211fc46d7de4%2F4dc565ff-787e-40a5-9492-e57b71e32bcd%2Fwf1timm_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Exercise 1. Find the probability density function for the sum of four inde-
pendent uniform random variables.
This exercise takes a little effort. Nonetheless, one can cut down on the
drudge work by being a little clever.
• The density is four third degree polynomials spliced together.
• The third degree polynomial on [0, 1] is easy to find, and you can prob-
ably even guess what it is looking at the example leading up to this
exercise.
• The density, as well as its first and second derivatives are continuous
at z = 1; note also that the derivative at z = 2 is zero. On [1,2], the
density has the form h(z) = az° +bz² +cz + d; and the four conditions
noted allow one to set up four linear equations in the four coefficients,
a, b, c, d.
• The density is clearly symmetric about the mean, so once ong finds the
density on [0, 2], finding the density on [2, 4] is immediate. One just has
to think out what the algebraic transformation implied by symmetry
must be. Again, looking at the previous results should help.
A graph of the density is presented in Figure 3. Superimposed on the graph
is a graph of the normal density with the same mean and variance: µ =
+++ = 2 and o? = + ++++
%3D
12
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