Given the PMF pN(n)=12⋅n⋅2−n for n=1,2,…
Find an expression for the conditional PMF of K given N, pK|N(k|n).
To find the conditional PMF of K given N, pK|N(k|n), we can use the definition of conditional probability.
The conditional PMF of K given N=k is given by:
pK|N(k|n) = P(K=k|N=n)
To find P(K=k|N=n), we need to calculate the probability that K=k when N=n.
The PMF pN(n) is given as:
pN(n) = 12⋅n⋅2^(-n), for n=1,2,...
We can calculate the denominator of the conditional probability by summing up the values of pN(n) for all possible n values:
P(N=n) = ΣpN(n) = Σ(12⋅n⋅2^(-n)), for n=1,2,...
Once we have calculated P(N=n), we can calculate the numerator of the conditional probability:
P(K=k and N=n) = P(K=k, N=n).
To find P(K=k, N=n), we need to consider the joint probability of K=k and N=n.
The joint PMF pK,N(k,n) is given by:
pK,N(k,n) = P(K=k, N=n)
Since we have the PMF pN(n), we can use it to calculate the joint PMF as follows:
pK,N(k,n) = pK|N(k|n) * pN(n)
Simplifying the expression, we have:
pK|N(k|n) = pK,N(k,n) / pN(n)
Therefore, the expression for the conditional PMF of K given N, pK|N(k|n), is given by:
pK|N(k|n) = pK,N(k,n) / pN(n)
Substituting the corresponding probabilities, we have:
pK|N(k|n) = (P(K=k, N=n)) / (P(N=n))
You can calculate the numerator by determining the joint probability of K=k and N=n, and you can calculate the denominator by summing up the values of pN(n) for all possible n.
To find the conditional PMF of K given N, pK|N(k|n), we need to use the definition of conditional probability. The conditional PMF is defined as the probability of K=k given that N=n.
Let's denote the conditional PMF as pK|N(k|n). The probability of K=k given N=n can be calculated using the formula:
pK|N(k|n) = P(K=k and N=n) / P(N=n)
To find the joint probability P(K=k and N=n), we can use the probability mass function (PMF) of K and N.
Given that the PMF of N is pN(n) = 12 * n * 2^(-n), we can express the joint probability P(K=k and N=n) as:
P(K=k and N=n) = P(K=k, N=n)
= P(N=n) * P(K=k|N=n)
= pN(n) * pK|N(k|n)
Therefore, we can rewrite the conditional PMF as:
pK|N(k|n) = (pN(n) * pK|N(k|n)) / pN(n)
Substituting the given PMF pN(n) = 12 * n * 2^(-n), we have:
pK|N(k|n) = (12 * n * 2^(-n) * pK|N(k|n)) / (12 * n * 2^(-n))
Simplifying the expression, we obtain:
pK|N(k|n) = pK|N(k|n)
Therefore, the conditional PMF of K given N, pK|N(k|n), is equal to the marginal PMF pK(k) and does not depend on N.