Moment generating function

In summary, the task at hand is to use conditional expectation to compute the moment generating function of a random variable Z, where Z is defined as the product of two other random variables X and T. The approach involves rewriting the moment generating function in terms of the conditional expectation and using properties of conditional expectation to simplify the expression. Ultimately, the moment generating function is found to be exp(10s/T), where T is a constant and X follows a uniform distribution with parameters 0 and 10.
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Homework Statement



Use conditional expectation to compute the moment generating function M_z(s) of the random variable Z=XT.


Homework Equations



X ~ R(0,10)
T ~ exp(0.1)

The Attempt at a Solution



By definition:

M_z(s) = E(exp(sZ))
=E(exp(sXT))

The only thing I can think of doing is using double expectation.

So M_z(s) = E[E(exp(sXT|X)] = E[E(exp(sXT|T)]...(1)

But here I'm stuck. Both X and T are continuous random variables so I can't just plug in the values of X and T one by one. Writing (1) as an integral of an expectation doesn't help either because I don't have an expression for the integrand and so cannot evaluate the integral. This question is really doing me in and any help to relieve my great pain would be much appreciated.
 
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  • #2


Hello,

Thank you for your forum post. It seems like you are on the right track with using double expectation. However, there is another approach that may be easier to follow.

First, we can rewrite the moment generating function as:

M_z(s) = E[exp(sXT)] = E[exp(sX) | T] (since Z = XT)

Next, we can use the properties of conditional expectation to rewrite this as:

M_z(s) = E[E[exp(sX) | T]] = E[M_x(s) | T] (since X ~ R(0,10))

Now, since T ~ exp(0.1), we know that the conditional distribution of X given T is also an exponential distribution with mean 10/T. Therefore, we can rewrite the inner expectation as:

E[exp(sX) | T] = M_x(s/T) (since X ~ R(0,10))

Finally, plugging this into our previous expression, we get:

M_z(s) = E[M_x(s/T) | T] (since X ~ R(0,10))

Now, we can use the law of total expectation to evaluate this expression:

M_z(s) = E[M_x(s/T) | T] = E[M_x(s/T)] = M_x(s/T) (since T is a constant)

Therefore, the moment generating function of Z is given by:

M_z(s) = M_x(s/T) = exp(10s/T) (since X ~ R(0,10))

I hope this helps. Let me know if you have any further questions.
 

What is a moment generating function?

A moment generating function (MGF) is a mathematical function used in probability theory and statistics to characterize the properties of a random variable. It is used to calculate moments of a probability distribution, such as mean, variance, and higher moments.

What is the purpose of a moment generating function?

The moment generating function allows us to easily calculate the moments of a probability distribution, which provide important information about the distribution, such as its shape, spread, and central tendency. It also allows us to derive other useful properties, such as the moment generating function of a linear combination of random variables.

How do you calculate a moment generating function?

The moment generating function of a random variable X is defined as M(t) = E[e^(tX)], where E represents the expected value operator. In other words, it is the expected value of the exponential function of the random variable X. To calculate the moment generating function, you need to know the probability distribution of X and use it to calculate the expected value.

What is the relationship between a moment generating function and a probability distribution?

The moment generating function uniquely determines the probability distribution of a random variable. This means that if two random variables have the same moment generating function, they also have the same probability distribution. However, the reverse is not always true - two random variables with the same probability distribution may have different moment generating functions.

Why is the moment generating function important in statistical analysis?

The moment generating function is important in statistical analysis because it provides a way to summarize and analyze the properties of a probability distribution in a compact and convenient form. It also allows us to easily calculate moments and other useful properties of a distribution, which are essential for many statistical procedures and techniques.

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