If random variable x has density f and random variable y has density g, then
random variable t = x + y has density
h(t) = {the integral from zero to t of} f(t-lambda) times g(lambda) d(lambda).
(i.e. the density of the sum is the convolution of the densities)
At 18:35 -0700 07/02/2000, David A. Heiser wrote:
>First Gautam Sethi used the term "convolution" for the product to
>two (uniform) densities. Aniko responded with a definition of
>convolution as the sum of two random variables. Then Jan de Leeuw
>stated that "convolution is the distribution of the sum". Herman
>Rubin stated that "convolution is the distribution of the sum". The
>idea that "convolution" represents the distribution of the sum of
>two distributions continues in the additional E mails.
>
>Everyone here is talking about "convolution" in an entirely
>different sense than what it is used in engineering. That is why I
>am confused.
>
>The convolution or Faltung integral is defined as follows: (Re: any
>advanced engineering mathematics text. Microsoft Outlook Express
>only transmits plain text or rich text, and therefore I cannot give
>the expression symbolically.)
>
>y(t) = {the integral from zero to t of} f(t-lambda) times g(lambda) d(lambda).
> Where t is a positive, finite time value.
> f is a function of time, describing an input (re: an
>x characteristic)
> g is a function of time, describing an fixed,
>operator on f (re: a parameter characteristic)
> y is the result, or output as a function of time.
> lambda is a variable of integration, with units of time.
>
>Taking the Laplace transforms of the above equation we have
>L(y) = L(f) times L(g)
>
>We can also take Fourier transforms.
>
>The convolution integral is the fundamental mathematical description
>of all control systems from aircraft flight to industrial processes.
>With current numerical computer methods, the above equations can be
>solved for any process.
>
>I tried this back in 1962. I tried defining the time statistical
>variations of the flow rate of solid oxidizer into a continuous
>mixing process by means of power spectral density. Given the
>transfer function of the mixing machine (obtained by pulse
>measurement methods), I would be able to arrive at estimates of the
>statistical variations in mixed propellant composition. I didn't get
>it finished, because I didn't have the computer tools and the
>knowledge at the time. This was also beyond the understanding of
>management and of the customer, and there was no interest to pursue
>it.
>
>Now when you are referring to "convolution" as a sum of
>distributions, how does it fit in?
>
>DAHeiser
>
--
===
Jan de Leeuw; Professor and Chair, UCLA Department of Statistics;
US mail: 8142 Math Sciences Bldg, Box 951554, Los Angeles, CA 90095-1554
phone (310)-825-9550; fax (310)-206-5658; email: [EMAIL PROTECTED]
http://www.stat.ucla.edu/~deleeuw and http://home1.gte.net/datamine/
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