Marginal Distribution of X & Conditional Density of λ in STAT134/150

In summary, the marginal distribution of X is a Poisson distribution with mean \lambda, and the conditional density of \lambda given X = x is an exponential distribution with mean x/\theta.
  • #1
ryzeg
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STAT134/150: Marginal distr. of a random var. w/ random var. param., given distrs

Hello,

I am a little shaky on my probability, so bear with me if this is a dumb question...

Anyway, the distributions of the two random variables are given:

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

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
The marginal distribution of X is a Poisson distribution with mean \lambda. The conditional density of \lambda given X = x is an exponential distribution with mean x/\theta.
 

What is the marginal distribution of X in STAT134/150?

The marginal distribution of X in STAT134/150 refers to the probability distribution of the random variable X, which represents the values of a particular variable in a given dataset. It is typically represented as a histogram or a probability density function (PDF), and it shows the frequencies or probabilities of each value of X occurring.

What is the conditional density of λ in STAT134/150?

The conditional density of λ in STAT134/150 refers to the probability distribution of the parameter λ, which represents the rate of occurrence of a particular event in a given dataset. It is typically represented as a function of λ, and it shows the probabilities of different values of λ given certain conditions or constraints.

How are the marginal distribution of X and the conditional density of λ related in STAT134/150?

The marginal distribution of X and the conditional density of λ are related in STAT134/150 through the concept of conditional probability. The conditional density of λ is the probability of λ occurring given a certain value of X, while the marginal distribution of X shows the overall frequencies or probabilities of X regardless of the value of λ. Together, they provide a more complete understanding of the relationship between these two variables in a dataset.

How can the marginal distribution of X and the conditional density of λ be visualized in STAT134/150?

The marginal distribution of X and the conditional density of λ can be visualized in STAT134/150 using various statistical tools such as histograms, box plots, and probability density functions. These visualizations help to illustrate the patterns and relationships between these two variables and provide a better understanding of the data.

Why is understanding the marginal distribution of X and the conditional density of λ important in STAT134/150?

Understanding the marginal distribution of X and the conditional density of λ is important in STAT134/150 because it allows for a deeper analysis and interpretation of the data. These concepts help to identify patterns and relationships between variables, which can inform decision-making and further statistical analysis. Additionally, understanding these distributions can aid in the development of predictive models and making accurate predictions about the data.

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