Sums of Independent Random Variables

In summary, the pdf for independent random variables can be found using convolution: f[sub:X+Y](a) = ∫ f[sub:X](a-y)f[sub:Y](y)dy. It is also possible to use the density function for a function of a random variable, given by f[sub:X+Y](x[u,v], y[u,v])(Jacobian Inverse(x[u,v], y[u,v])), and integrate it. The Jacobian determinant, (∂u/∂x)(∂v/∂y)-(∂u/∂y)(∂v/∂x), is also relevant. However, when finding the density function for U=X+Y, it is important to know
  • #1
jrk012
7
0

Homework Statement



So, I know the pdf for independent random variables is found by using the convolution; the pdf is f[sub:X+Y](a) = ∫ f[sub:X](a-y)f[sub:Y](y)dy, but can I just use the density function for a function of a random variable instead; that is: f[sub:X+Y](x[u,v], y[u,v])(Jacobian Inverse(x[u,v], y[u,v])) and then integrate it? It seems much easier that way.


Homework Equations



f[sub:X+Y](a) = ∫ f[sub:X](a-y)f[sub:Y](y)dy

f[sub:X+Y](x[u,v], y[u,v])(Jacobian Inverse(x[u,v], y[u,v]))

Jacobian Determinant: (∂u/∂x)(∂v/∂y)-(∂u/∂y)(∂v/∂x)


The Attempt at a Solution



More of a question on coursework than homework or a specific problem. However when I find the density function when U=X+Y I get f(x[u-v], y[v])
 
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  • #2
jrk012 said:

Homework Statement



So, I know the pdf for independent random variables is found by using the convolution; the pdf is f[sub:X+Y](a) = ∫ f[sub:X](a-y)f[sub:Y](y)dy, but can I just use the density function for a function of a random variable instead; that is: f[sub:X+Y](x[u,v], y[u,v])(Jacobian Inverse(x[u,v], y[u,v])) and then integrate it? It seems much easier that way.


Homework Equations



f[sub:X+Y](a) = ∫ f[sub:X](a-y)f[sub:Y](y)dy

f[sub:X+Y](x[u,v], y[u,v])(Jacobian Inverse(x[u,v], y[u,v]))

Jacobian Determinant: (∂u/∂x)(∂v/∂y)-(∂u/∂y)(∂v/∂x)


The Attempt at a Solution



More of a question on coursework than homework or a specific problem. However when I find the density function when U=X+Y I get f(x[u-v], y[v])

So U = X+Y, but what is V?

RGV
 
  • #3
They said "Hint: V=Y" so I wasn't sure if that was supposed to be assumed known or if there were different ways to do it. Sorry I forgot to put that in there!
 

FAQ: Sums of Independent Random Variables

What are independent random variables?

Independent random variables are variables that do not influence each other in any way and have no correlation. This means that the value of one variable does not affect the value of the other variable.

What is the significance of studying sums of independent random variables?

Studying sums of independent random variables is important in many fields, such as statistics and probability, as it allows us to make predictions and analyze data in a meaningful way. It also helps in understanding complex systems and their behavior.

How do you calculate the sum of independent random variables?

To calculate the sum of independent random variables, you simply add the values of each variable together. This is because independent variables do not affect each other, so their values can be added without any adjustment.

What is the central limit theorem and how does it relate to sums of independent random variables?

The central limit theorem states that the sum of a large number of independent random variables will follow a normal distribution, regardless of the distribution of the individual variables. This is significant in studying sums of independent random variables as it allows us to make predictions and perform statistical analyses.

Can you give an example of sums of independent random variables in real life?

One example of sums of independent random variables in real life is the rolling of multiple dice. Each die represents an independent random variable, and the sum of the values on each die follows a normal distribution according to the central limit theorem.

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