Suppose X_n is a sequence of independent Bernoulli random variables and p(X_n=1)=p_n. If Y=∑_(n=1)^∞▒X_n is convergent with probability 1 ,is E(Y) convergent?

To determine whether the expected value E(Y) is convergent, we need to examine the sequence of probabilities p_n and determine if it satisfies certain conditions.

First, note that E(Y) can be written as the infinite sum of E(X_n) for n ranging from 1 to infinity. This is due to the linearity of the expected value operator.

E(Y) = E(∑_(n=1)^∞▒X_n) = ∑_(n=1)^∞▒E(X_n)

Now, if E(Y) is convergent, that means the series ∑_(n=1)^∞▒E(X_n) is convergent.

For each X_n, we know that E(X_n) = p_n. Therefore, the convergence of E(Y) depends on the convergence of the series ∑_(n=1)^∞▒p_n.

If the series ∑_(n=1)^∞▒p_n converges, then E(Y) is convergent.

However, we cannot conclude the convergence of E(Y) solely based on the convergence of the series ∑_(n=1)^∞▒p_n. It is possible to have a convergent series ∑_(n=1)^∞▒p_n, but E(Y) is still divergent.

Therefore, we cannot determine whether E(Y) is convergent solely based on the given information. Additional information about the probabilities p_n or the series ∑_(n=1)^∞▒p_n would be required to make a conclusion.