Marginal Distribution of X w/ Lambda Parameter: Probability Help

In summary, the conversation discusses finding the marginal distribution of X and the conditional density for λ given a value for X, with the use of Bayes' rule. The speaker mentions having equations for dependent distributions but is unsure how to apply them to the problem. They also mention difficulty with evaluating an integral using Mathematica and a TI-89T calculator.
  • #1
ryzeg
9
0
I am a little shaky on my probability, so bear with me if this is a dumb question...

Anyway, these two random variables are given:

X : Poisson ([tex]\lambda[/tex])
[tex]\lambda[/tex] : Exponential ([tex]\theta[/tex])

And I simply need the marginal distribution of X and the conditional density for [tex]\lambda[/tex] given a value for X

I have all the equations for dependent distributions, but do not know how to apply them to this ostensibly easy problem...

Any help?
 
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  • #2


You'll need Bayes' rule for this. What results have you got so far?
 
  • #3


I was doing this, but I think it is wrong:

[tex]
f_X(x) = \int^{\lambda=\infty}_{\lambda=0} \frac{\lambda^{x}}{x!} e^{-\lambda} \times \theta e^{-\theta \lambda} d \lambda
[/tex]

Plugging this integral into Mathematica gives a really nasty output with a incomplete gamma function, and my TI-89T cannot evaluate it.
 
  • #4


I take that back; the integral is doable with a little manipulation. Damn machines...
 

1. What is the definition of marginal distribution of X with lambda parameter?

The marginal distribution of X with lambda parameter is a statistical concept that describes the probability distribution of a random variable X, given a specific value of the parameter lambda. It is calculated by summing or integrating over all possible values of the other variables in the joint distribution.

2. How is the marginal distribution of X with lambda parameter different from the joint distribution?

The joint distribution describes the probability of two or more random variables occurring together, while the marginal distribution of X with lambda parameter focuses on the probability of a single random variable X occurring with a specific value of lambda, regardless of the other variables in the joint distribution.

3. What is the significance of the lambda parameter in the marginal distribution of X?

The lambda parameter is a constant that determines the shape and scale of the marginal distribution of X. It can affect the mean, variance, and other important characteristics of the distribution, making it a crucial factor in statistical analysis and modeling.

4. How can the lambda parameter be estimated from a sample of data?

The lambda parameter can be estimated using various statistical techniques, such as maximum likelihood estimation or method of moments. These methods involve using the data to calculate the most likely value of lambda that would produce the observed sample data.

5. What are some real-world applications of the marginal distribution of X with lambda parameter?

The marginal distribution of X with lambda parameter is used in a wide range of fields, including finance, economics, biology, and engineering. It can be applied to model various phenomena, such as stock prices, population growth, and failure rates of mechanical components.

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