Question

Consider the following discrete time process: at integer times, a marble arrives which can be one of m colors. Each color is equally likely; all events at different times are independent. As each marble arrives, it is placed into a bowl Unless there is a marble of that color already in the bowl. Then, that marble is removed from the bowl and the new marble does not go into the bowl. (The pair is placed in a separate bin for shipping). Let Gn denote the number of marbles currently in the bowl after the nth marble arrives. Compute,(a) the expectation E(Gn).(b) limn→[infinity] E(Gn).

Answer 1

Below are the responses to the given question:

Explanation:

Let X become the random marble variable & g have been any function.

Now.

For point a:

When X is discreet, the g(X) expectation is defined as follows

Then there will be a change of position.

E[g(X)] = X x∈X g(x)f(x)

If f is X and X's mass likelihood function support X.

For point b:

When X is continuing the g(X) expectations is calculated as, E[g(X)] = Z ∞ −∞ g(x)f(x) dx, where f is the X transportation distances of probability.If E(X) = −∞ or E(X) = ∞ (i.e., E(|X|) = ∞), they say it has nothing to expect from EX is occasionally written to stress that a specific probability distribution X is expected.Its expectation is given in the form of,E[g(X)] = Z x −∞ g(x) dF(x). , sometimes for the continuous random vary (x). Here F(x) is X's distributed feature. The anticipation operator bears the lineage of comprehensive & integral features. The superposition principle shows in detail how expectation maintains equality and is a skill.