Question

Question about using the convolution of distribution:

1. we have the formula: integral fx(x)fy(z-x)dx=integral fx(z-x)fy(x)dx

I know this are equivalent. However, how do I decide which side I should use ?

For example,X~Exp(1) and Y~Unif [0,1] X and Y independnt and the textbook use fx(z-x)fy(x)dx.

However, can I use the left hand side fx(x)fy(z-x)dx???is there any constraint for using left or right or actually both can lead me to the right answer???

2. For X and Y are independent and exponential distribution, however, why all the text book just ignore the infinity?

they use integral from 0 to **z** fx(x)*fy(z-x) to
find the distribution X+Y, however, in my understand exponential
distribution should be from 0 to infinity

Answer #1

Convolution of distribution integral range
1.Example: Exponential distribution convolution
fx+y(z)=intergral from 0 to z fx(x)fy(z-x))dx
how we get the range 0 to z???
Better to use mathematics and a graph to explain.
Also, when it ask find the density function X+Y, should I just
add two exponential density function together?

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variable Y is related to X via Y = (-ln(1 - X))^1/3.
(a) Demonstrate that the pdf of Y is fY (y) = 3y^2 e^-y^3,
y>0. (Hint: Work out FY (y))
(b) Determine E[Y ]. (Hint: Use Wolfram Alpha to undertake the
integration.)

Derive the following using all known inferences rules and
equivalences
including QE.
Remember that equivalences afford you greater power than
derivation rules, because
you are permtted to substitute equivalent sub formlas within a wff.
For example, all the
moves on the left below are legitimate inferences even though the
first conditional is the
main operator. However, it is important to remember that you must
always be operating
on a true self-contained wff. The moves on the right are not
legitimate...

(1) Using a generator for a binomial distribution, we will test
the results of Example 3.8.2. Using software generate 500 random
deviates for X from a B(10, 0.3) distribution and 500 random
deviates for Y from a B(5, 0.3) distribution. Add corresponding
random deviates from each distribution to form an empirical W=X+Y.
Then use the theoretical result of Example 3.8.2 and directly
generate another 500 random deviates for W from a B(15, 0.3). Order
the result of the sum of...

If a random variable X has a beta distribution, its probability
density function is
fX (x) = 1 xα−1(1 − x)β−1 B(α,β)
for x between 0 and 1 inclusive. The pdf is zero outside of
[0,1]. The B() in the denominator is the beta function, given by
beta(a,b) in R.
Write your own version of dbeta() using the beta pdf formula
given above. Call your function mydbeta(). Your function can be
simpler than dbeta(): use only three arguments (x, shape1,...

Problem #1 Confidence Interval for Means using the t and
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the percent tip at a restaurant when a message indicating that the
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tips from a random sample of patrons who received such a bill,
measured in percent of the total bill:
20.8 18.7
19.9 20.6
21.9 23.4
22.8 24.9
22.2 20.3
24.9 22.3
27.0 20.4
22.2 24.0
21.1 22.1
22.0 22.7
Open an...

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