Density function of product of random variables

In summary, the conversation discusses forming a new random variable Z as the product of two independent random variables X and Y. The question is posed about the density function of Z, given the density functions of X and Y (f(x) and f(y) respectively). The possibility of using logs and convolution is mentioned, but it is suggested to instead start by considering the probability of Z being less than or equal to a certain value, using the conditional probability of XY being less than or equal to that value given Y.
  • #1
khotsofalang
21
0
suppose you have two random variables X and Y which are independent,
we want to form a new random variable Z=XY, if f(x) and f(y) are density functions
of X and Y respectively what is the density function of Z?

I tried taking logs and applying convolution, but it did not really work
 
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  • #2
It isn't clear to me from your writing whether X and Y have the same distribution - you used the same name for their densities - but think about this start: it will work whether they are identically distributed or not. Start with this
[tex]
P(Z \le z) = E_Y \left[P\left( XY \le z \mid Y = y\right) \right]
[/tex]

and see what develops.
 

1. What is a density function?

A density function is a mathematical function that describes the probability distribution of a continuous random variable. It represents the relative likelihood that the variable takes on a specific value or falls within a certain range of values.

2. How is the density function of product of random variables calculated?

The density function of the product of two random variables is calculated by taking the product of their individual density functions. For example, if X and Y are two random variables, the density function of their product XY is given by fXY(x,y) = fX(x) * fY(y).

3. What is the relationship between the density function and the probability distribution function?

The density function is the derivative of the probability distribution function. It represents the rate of change of the probability distribution at a given point, while the probability distribution function gives the probability of a random variable taking on a specific value or falling within a certain range of values.

4. Can the density function of product of random variables be negative?

No, the density function of product of random variables cannot be negative. Since it represents the probability distribution, it must always be non-negative. If the calculated density function is negative, it means there is an error in the calculation.

5. How is the density function of product of independent random variables different from that of dependent random variables?

The density function of product of independent random variables is simply the product of their individual density functions, while for dependent random variables, it takes into account the relationship between the variables. In other words, the density function of dependent random variables considers the joint probability distribution, while for independent random variables, the joint probability is simply the product of their individual probabilities.

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