(a)Determine the moment generating function Y=2x, i.e MY(t)
(b) Compute E(√(2x))

Question

(a)Determine the moment generating function Y=2x, i.e MY(t)(b) Compute E(√(2x))

alenahelenash · Accepted Answer

To determine the moment generating function (MGF) of a random variable Y, we use the formula:

M Y (t) = E (e^{t Y})

In this case, we have Y = 2X, where X is some other random variable. Therefore, we can write:

M Y (t) = E (e^{t (2 X)})

We do not have any information about the distribution of X, so we cannot calculate the MGF directly. However, we can use a property of MGFs to derive the MGF of Y. Specifically, if X and Y are independent random variables, then the MGF of their sum is equal to the product of their MGFs. That is:

M_{X + Y} (t) = M_{X} (t) M_{Y} (t)

In this case, we have Y = 2X, so we can write:

M_{X + Y} (t) = M_{X} (t) M_{2 X} (t)

Now, we need to find the MGF of 2X. We can use the same formula as before:

M_{2 X} (t) = E (e^{t (2 X)})

Again, we do not have any information about the distribution of X, so we cannot calculate the MGF directly. However, we can use another property of MGFs to derive the MGF of 2X. Specifically, if c is a constant, then the MGF of cX is equal to the MGF of X raised to the power of c. That is: