Let Y be the discrete random variable with probability function
p(y) = (1/2)^(y+1) y = 0, 1, 2,...
Find the moment-generating function m(t) for Y.
To find the moment-generating function (MGF) for the random variable Y, we need to use the formula for the MGF, which is defined as:
m(t) = E(e^(tY))
Where E() denotes the expected value and t is a parameter. Let's find the MGF step by step:
Step 1: Determine the range of the random variable Y
In this case, the random variable Y can take on values from 0 to infinity (y = 0, 1, 2, ...)
Step 2: Write the definition of the MGF
m(t) = E(e^(tY))
Step 3: Calculate the expected value E(Y)
To calculate the expected value, we need to use the probability function p(y) and sum over all possible values of Y:
E(Y) = Σ(y * p(y)), where Σ represents the sum over all possible values of Y.
p(y) = (1/2)^(y+1) for y = 0, 1, 2, ...
So the expected value becomes:
E(Y) = Σ(y * (1/2)^(y+1))
Step 4: Simplify the expression for the expected value
We can use the formula for the sum of an infinite geometric series to simplify the expected value:
Σ(y * (1/2)^(y+1)) = Σ(y * (1/2) * (1/2)^y)
= (1/2) * Σ(y * (1/2)^y)
To find the sum of the infinite geometric series, we know that for |r| < 1, the sum is given by:
Σ(n * r^n) = r / (1 - r)^2
In our case, r = 1/2, so:
Σ(y * (1/2)^y) = (1/2) / ((1/2) - 1)^2
= (1/2) / (-1/2)^2
= 1 / (1/4)
= 4
Therefore, E(Y) = 4.
Step 5: Evaluate the MGF
Now that we have the expected value, we can substitute it back into the MGF formula:
m(t) = E(e^(tY))
= Σ(e^(ty) * p(y))
= Σ(e^(ty) * (1/2)^(y+1))
Substituting p(y) = (1/2)^(y+1) and simplifying, we have:
m(t) = Σ(e^(ty) * (1/2)^(y+1))
= Σ((e^t/2)^(y+1))
= (e^t/2) * Σ((e^t/2)^y)
Again using the formula for the sum of an infinite geometric series, we get:
Σ((e^t/2)^y) = 1 / (1 - e^t/2), where |e^t/2| < 1
Therefore, the MGF for the random variable Y is:
m(t) = (e^t/2) / (1 - e^t/2)
This is the moment-generating function for Y.