Marginal PMG of of 2 random variables with Joint PMF

In summary, marginal PMF, or marginal probability mass function, is the probability distribution of a single random variable in a joint probability distribution. It is calculated by summing up the probabilities of all possible outcomes for that variable. Marginal PMF and joint PMF are related, but the joint PMF can be used to calculate the marginal PMF while the reverse is not true. Marginal PMF is useful in statistical analysis as it allows us to study the behavior of a single variable in a system. It can also be used to calculate conditional probabilities.
  • #1
whitejac
169
0

Homework Statement


Consider two random variables X and Y with joint PMF given by:
PXY(k,L) = 1/(2k+l), for k,l = 1,2,3,...

A) Show that X and Y are independent and find the marginal PMFs of X and Y
B) Find P(X2 + Y2 ≤ 10)

Homework Equations


P(A)∩P(B)/P(B) = P(A|B)
P(A|B) = P(A) if independent

The Attempt at a Solution


Choosing two arbitrary numbers to show P(A|B) = P(A)

P(x<2) ∩ P(y≤1) / P(y≤1)
=P(1,1) + P(1,2) + P(1,3)... + P(1, L) ∩ P(1,1) + P(1,2) + P(1,3)...+ P(k,1) / P(y≤1)
=P(1,1) / P(1,1) + P(1,2) + P(1,3)... + P(k,1)
Note: Geometric series
=(1/4) / (1/2)(1/2^k)
=(1/4) / (1/4)(1 / (1 - (1/2)))
=(1/4) / 2
= 1/2

P(A) = P(x<2) = P(1,L) =
=1/4 (1/2L)
=1/4 (1 / (1 - (1/2)))
=1/4 (2)
= 1/2

X and Y are independent.

How does one find the Marginal PMF of this equation then? The ones I've seen before in discrete sections were pre-made and finite... meaning that they were a table of results for X =1,2,3... and Y = 1,2,3... for the range of each. Should I be finding a summation for each value of X + Y?

I haven't considered part B yet.
 
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  • #2
whitejac said:

Homework Statement


Consider two random variables X and Y with joint PMF given by:
PXY(k,L) = 1/(2k+l), for k,l = 1,2,3,...

A) Show that X and Y are independent and find the marginal PMFs of X and Y
B) Find P(X2 + Y2 ≤ 10)

Homework Equations


P(A)∩P(B)/P(B) = P(A|B)
P(A|B) = P(A) if independent

The Attempt at a Solution


Choosing two arbitrary numbers to show P(A|B) = P(A)

P(x<2) ∩ P(y≤1) / P(y≤1)
=P(1,1) + P(1,2) + P(1,3)... + P(1, L) ∩ P(1,1) + P(1,2) + P(1,3)...+ P(k,1) / P(y≤1)
=P(1,1) / P(1,1) + P(1,2) + P(1,3)... + P(k,1)
Note: Geometric series
=(1/4) / (1/2)(1/2^k)
=(1/4) / (1/4)(1 / (1 - (1/2)))
=(1/4) / 2
= 1/2

P(A) = P(x<2) = P(1,L) =
=1/4 (1/2L)
=1/4 (1 / (1 - (1/2)))
=1/4 (2)
= 1/2

X and Y are independent.

How does one find the Marginal PMF of this equation then? The ones I've seen before in discrete sections were pre-made and finite... meaning that they were a table of results for X =1,2,3... and Y = 1,2,3... for the range of each. Should I be finding a summation for each value of X + Y?

I haven't considered part B yet.

The marginal pmf of ##X## is ##P_X(k) = P(X = k) = P(X = k \: \& \; Y= \text{anything})##. Note that for ##l = 1,2,3 \ldots## the different events ##\{ X = k, Y = l \}## are disjoint (mutually exclusive), so the probability of ##\{ X = k \: \& \; Y \in \{ 1,2,3, \ldots \} \} ## is just the sum of their probabilities for different ##l##.
 
  • #3
Ray Vickson said:
The marginal pmf of ##X## is ##P_X(k) = P(X = k) = P(X = k \: \& \; Y= \text{anything})##. Note that for ##l = 1,2,3 \ldots## the different events ##\{ X = k, Y = l \}## are disjoint (mutually exclusive), so the probability of ##\{ X = k \: \& \; Y \in \{ 1,2,3, \ldots \} \} ## is just the sum of their probabilities for different ##l##.
How does this differ from PXY, because that's what it sounds like. If the marginal pmf is P(X = k and Y equals anything), it sounds like I'm looking for PXY(k,l)
 
  • #4
whitejac said:
How does this differ from PXY, because that's what it sounds like. If the marginal pmf is P(X = k and Y equals anything), it sounds like I'm looking for PXY(k,l)
Sorry, it differs by definition. PXY(K,L) = PX(K)PY(L) which, in this case, equals:
1 / (2K) * 1 / (2L).
 
  • #5
whitejac said:
Sorry, it differs by definition. PXY(K,L) = PX(K)PY(L) which, in this case, equals:
1 / (2K) * 1 / (2L).

Right: and ##P_X(k) = \sum_{l=1}^{\infty} P_{XY}(k,l)##, etc.

By the way, the fact that ##P_{XY}(k,l) = P_X(k) P_Y(l)## is what makes ##X## and ##Y## independent. You did not convincingly show independence in your previous post, because you only showed a few examples of ##P(X \in A \; \& \; Y \in B) = P(X \in A) \, P(Y \in B)## for one or two special cases of ##A## and ##B##. To prove independence, you need to establish this for all possible choices of ##A## and ##B##.
 

Related to Marginal PMG of of 2 random variables with Joint PMF

What is the definition of marginal PMF?

Marginal PMF, or marginal probability mass function, is a statistical term used to describe the probability distribution of a single random variable in a joint probability distribution. It represents the probability of a single outcome of a random variable, without considering the other variables in the system.

How is marginal PMF calculated?

To calculate the marginal PMF of a single random variable in a joint probability distribution, you need to sum up the probabilities of all possible outcomes for that variable. This can be done by either using a table or by using the formula for marginal probability mass function: P(X=x) = ∑P(X=x,Y=y), where X and Y are the two random variables in the joint distribution.

What is the relationship between marginal PMF and joint PMF?

Marginal PMF and joint PMF are related as follows: the marginal PMF is the probability distribution of a single random variable in a joint distribution, while the joint PMF represents the probability of a combination of outcomes for two or more random variables. The joint PMF can be used to calculate the marginal PMF, but the reverse is not true.

Why is marginal PMF useful in statistical analysis?

Marginal PMF is useful in statistical analysis because it allows us to study the behavior of a single random variable in a system, without considering the other variables. This can help in understanding the overall behavior of the system and making predictions about future outcomes.

Can marginal PMF be used to calculate conditional probabilities?

Yes, marginal PMF can be used to calculate conditional probabilities. Conditional probabilities represent the probability of an event occurring given that another event has already occurred. By using marginal PMF, we can calculate the probability of a single variable given the outcome of another variable in a joint probability distribution.

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