Joint distribution function of z and |z|

In summary, the joint distribution function of z and |z| is a statistical measure that describes the probability of a random variable z and its absolute value |z| occurring simultaneously. It is calculated by integrating the probability density function (PDF) over all possible values of z and |z|, and provides information about the dependence or independence of these two variables. This function can also be used to calculate conditional probabilities and has various real-world applications in fields such as finance, economics, and engineering.
  • #1
snazmn
1
0
X and Y are 2 independent gaussian random variables with parameter a.
Z = XY / (X-Y)
W = XY / |X-Y|
I am to find the joint distribution function of z and w.
I know how to find pdf of Z but how could I use it to find the joint distribution function of z and w?
 
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  • #2
Here Z and W have no joint pdf (because Z=+/-W) so you'll have to calculate the joint cdf directly. I'd suggest breaking it up into special cases, for example P[Z<=z,W<=w]=(1/2)P[Z<=z] if -z<=w<z.
 

FAQ: Joint distribution function of z and |z|

1. What is the joint distribution function of z and |z|?

The joint distribution function of z and |z| is a statistical measure that describes the probability of a random variable z and its absolute value |z| occurring simultaneously. This function is commonly used to analyze the relationship between two random variables and their respective probabilities.

2. How is the joint distribution function of z and |z| calculated?

The joint distribution function of z and |z| is calculated by taking the probability density function (PDF) of z and |z| and integrating it over all possible values of z and |z|. This integration results in a cumulative probability value for each pair of values, representing their joint probability.

3. What does the joint distribution function of z and |z| tell us about the relationship between these two variables?

The joint distribution function of z and |z| provides information about the dependence or independence of these two variables. If the function has a constant value, it suggests that z and |z| are independent, while a non-constant function indicates some level of dependence between them.

4. Can the joint distribution function of z and |z| be used to calculate conditional probabilities?

Yes, the joint distribution function of z and |z| can be used to calculate conditional probabilities. By dividing the joint probability by the marginal probability of z or |z|, we can obtain the conditional probability of z given |z| or vice versa.

5. What are some real-world applications of the joint distribution function of z and |z|?

The joint distribution function of z and |z| has various applications in fields such as finance, economics, and engineering. It can be used to analyze the relationship between two investment assets, determine the effectiveness of a marketing campaign, or model the relationship between two physical variables in a system.

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