Marginal Distribution
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Find the marginal PMF pK(k) as a function of k. For simplicity, provide the answer only for the case when k is an even number. (The formula for when k is odd would be slightly different, and you do not need to provide it).
Hint: You may find the following helpful: ∑i=0∞ri=11−r for 0<r<1.
For k=2,4,6,…:
pK(k)=
To find the marginal PMF pK(k) for the case when k is an even number, we need to consider a joint probability mass function (PMF) and use the concept of marginalization.
Let's assume we have a joint PMF pX,Y(x, y) for two random variables X and Y. To obtain the marginal PMF for X, we sum up the probabilities of all possible values of X while fixing the value of Y.
In this case, we need to find the marginal PMF pK(k) for the random variable K, where k is an even number.
To do this, we can use the following formula:
pK(k) = ∑ pX,Y(x, y)
where the summation is taken over all values of x and y that satisfy the condition k = x + y, and x and y are even numbers.
Using the hint provided, ∑i=0∞ri=11−r for 0<r<1, we can simplify the calculation.
For k = 2, we can express the summation as:
pK(2) = ∑ pX,Y(x, y)
where the summation is taken over all values of x and y that satisfy the condition 2 = x + y, and x and y are even numbers.
Since there is only one possibility, x = 0 and y = 2, the probability can be written as:
pK(2) = pX,Y(0, 2)
Similarly, for k = 4, we have:
pK(4) = ∑ pX,Y(x, y)
where the summation is taken over all values of x and y that satisfy the condition 4 = x + y, and x and y are even numbers.
Since there are two possible combinations, x = 0, y = 4 and x = 2, y = 2, the probabilities can be written as:
pK(4) = pX,Y(0, 4) + pX,Y(2, 2)
Continue this pattern for each even value of k, and you will have the marginal PMF pK(k) as a function of k.