How Do You Find the PDF of Z and Calculate Mean and Variance Using MGF?

[silkdigital]

Hi guys,

I'm really stuck on the following questions, not sure as to how to approach it:

Let X and Y be random variables for which the joint pdf is as follows:

f(x,y) = 2(x+y) for 0 <= x <= y <= 1
and 0 otherwise.

Find the pdf of Z = X + Y

And also:

Suppose that X is a random variable for which the mgf is as follows:

/u(t) = e^(t^2 + 3t) for minus infinity < t < infinity

Find the mean and variance for X.
I know that the answers are 3 and 2 respectively, but was unsure how they got to the answer, do I need to integrate by parts?

Any help would be appreciated! Thanks guys :)

 

[mathman]

Hi guys,

I'm really stuck on the following questions, not sure as to how to approach it:

Let X and Y be random variables for which the joint pdf is as follows:

f(x,y) = 2(x+y) for 0 <= x <= y <= 1
and 0 otherwise.

Find the pdf of Z = X + Y

And also:

Suppose that X is a random variable for which the mgf is as follows:

/u(t) = e^(t^2 + 3t) for minus infinity < t < infinity

Find the mean and variance for X.
I know that the answers are 3 and 2 respectively, but was unsure how they got to the answer, do I need to integrate by parts?

Any help would be appreciated! Thanks guys :)

I'll address the second question only. The moments are obtained from the moment generating function by simply taking derivatives and setting t = 0. As you must be aware, the variance is the second moment minus square of first moment.

 

[silkdigital]

Figured out second question now, pretty straightforward in hindsight. Any help on the first one? ;)

 

[chiro]

Figured out second question now, pretty straightforward in hindsight. Any help on the first one? ;)

Have you tried a transformation? Let u = x + y. Now use that transformation to get a integral in terms of u, take into account limits and then use transformation theorem to relate g(u) = 2(x+y) = 2u to another PDF f(u) which represents the distribution of Z.

 

[blue_raver22]

For the first question, to find the pdf of Z = X + Y, we can use the cumulative distribution function (CDF) approach.

Let Fz(z) be the CDF of Z. Then, we have:

Fz(z) = P(Z <= z) = P(X + Y <= z)

Since X and Y are non-negative and have a joint pdf defined over 0 <= x <= y <= 1, we can rewrite the above as:

Fz(z) = ∫∫ f(x,y) dx dy, where the limits of integration are 0 <= x <= min(z-y, 1) and y <= z-x

We can then solve this integral to get the CDF of Z. Taking the derivative of this with respect to z will give us the pdf of Z.

For the second question, to find the mean and variance of X, we can use the moment generating function (MGF) approach.

The MGF of X is defined as:

Mx(t) = E[e^(tX)]

To find the mean and variance, we can use the following formulas:

Mean = Mx'(0)
Variance = Mx''(0) - [Mx'(0)]^2

So, to find the mean, we can differentiate the MGF with respect to t and then evaluate it at t = 0. Similarly, to find the variance, we can differentiate the MGF twice with respect to t and then evaluate it at t = 0.

In this case, we have the MGF as e^(t^2 + 3t). Differentiating this once gives us 2te^(t^2 + 3t), and evaluating it at t = 0 gives us the mean as 3.

Similarly, differentiating the MGF twice gives us (4t + 6)e^(t^2 + 3t), and evaluating it at t = 0 gives us the variance as 2.

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