What is the PMF of a sum of two discrete random variables?

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In summary, the problem is to show that for any integer n≥2, the PMF P(X=k|X+Y=n) is uniform when X and Y are two independent discrete random variables with the same geometric pmf. The solution involves using the definition of conditional probability and realizing that P(X+Y=n) is the same as P(X=k, Y=n-k). With this understanding, the problem can be solved by finding the realization of Y given X=k and n.
  • #1
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Hi,

I'm working a problem and I'm stuck on one part. Consider, X and Y, two independent discrete random variables who have the same geometric pmf. Show that for all n ≥ 2, the PMF

P(X=k|X+Y=n) is uniform.

Now, this equals: P(X=k)P(Y=n-k)/P(X+Y=n), which follows from the definition of conditional probability. Since the X and Y have the same geometric pmf the numerator is easy to calculate, but I'm stuck on what exactly P(X+Y=n) is. I know it's the joint PMF, but how can I relate it to the problem (i.e. to the fact that X and Y have the same geo PMF, that X and Y are independent etc). Any help is appreciated.

Thanks,
Alex
 
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  • #2
To be more specific, the problem is to show for any integer n≥2
 
  • #3
Hey Alupsaiu and welcome to the forums.

What are finding difficult about the P(X + Y = n)?

You are given the realization of X (X = k), and you are given n, so based on that you should be able to get the realization of Y to figure out your probability.

That probability is just the probability that given n, it represents the probability that X = k and Y = n - k, In other words it is the same as saying P(X = k, Y = n - k).
 
  • #4
Hey, thanks for the reply. I figured the problem out a while ago, I don't know exactly why I found it confusing, long day I suppose haha. Thanks for the help though!
 
  • #5


Hi Alex,

The PMF (probability mass function) of a sum of two discrete random variables is a function that gives the probability of obtaining a particular sum of the two variables. In this case, we are looking at the PMF of X and Y, given that their sum is equal to n. This can be written as P(X+Y=n).

In order to calculate this, we can use the convolution formula for discrete random variables, which states that the PMF of the sum of two independent random variables is the sum of their individual PMFs. In other words, P(X+Y=n) = P(X=k)P(Y=n-k). This is because X and Y are independent, meaning that the outcome of one does not affect the outcome of the other.

Since X and Y both have the same geometric PMF, we can substitute in the formula for a geometric PMF, which is P(X=k) = p(1-p)^(k-1). Therefore, P(X+Y=n) = p(1-p)^(k-1) * p(1-p)^(n-k-1) = p^2(1-p)^(n-2).

Now, to show that the PMF P(X=k|X+Y=n) is uniform, we need to show that it is constant for all values of k. This means that for any value of k, the probability of X being equal to k given that X+Y=n is the same. Using the formula you provided, we can see that P(X=k|X+Y=n) = p(1-p)^(k-1) * p(1-p)^(n-k-1) / p^2(1-p)^(n-2) = (1-p)^(k-1).

This expression is independent of n, meaning that for any value of n, the probability of X being equal to k given that X+Y=n is the same. Therefore, the PMF P(X=k|X+Y=n) is indeed uniform.

I hope this helps! Don't hesitate to reach out if you have any further questions.

Best,
 

What is the PMF of a sum of two DRV?

The PMF (Probability Mass Function) of a sum of two DRV (Discrete Random Variables) is a mathematical function that describes the probabilities of all possible outcomes for the sum of two discrete random variables. It is used to determine the likelihood of a specific sum occurring in a given set of data.

What is the formula for calculating the PMF of a sum of two DRV?

The formula for calculating the PMF of a sum of two DRV is given by P(X+Y=k) = ∑P(X=i)P(Y=k-i), where X and Y are the two random variables, k is the sum of the two variables, and i is the index of summation.

What are the properties of the PMF of a sum of two DRV?

The PMF of a sum of two DRV has the following properties:

  • It is always non-negative, with a value between 0 and 1.
  • The sum of all probabilities is equal to 1.
  • The PMF is symmetric, meaning that P(X+Y=k) = P(Y+X=k).
  • The PMF is commutative, meaning that P(X+Y=k) = P(Y+X=k).

How is the PMF of a sum of two DRV related to the individual PMFs of the two variables?

The PMF of a sum of two DRV is related to the individual PMFs of the two variables through the convolution operation. This operation combines the individual PMFs to create the PMF of the sum of the two variables. It is represented by the formula P(X+Y=k) = ∑P(X=i)P(Y=k-i).

What are the common applications of the PMF of a sum of two DRV?

The PMF of a sum of two DRV has several applications in various fields, including:

  • Probability theory and statistics: It is used to determine the likelihood of a specific sum occurring in a given set of data.
  • Finance: It is used to model the risk and return of investment portfolios.
  • Engineering: It is used to study the reliability of systems and components.
  • Computer science: It is used in cryptography and coding theory.

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