Question

For two random variables X and Y, the joint moment generating function (jmgf) is defined as

Mx,y (t₁, t₂) = E[eᵗ¹ˣ⁺²ʸ].
Find the marginal moment generating functions for X and Y alone. (HINT: use strategic values of t₁ and t₂.)

Answer 1

The marginal moment generating functions for X and Y alone can be obtained from the joint moment generating function by setting the appropriate variables to zero.

Step-by-step explanation:

The marginal moment generating function for a single random variable provides information about the distribution of that variable alone. To find the marginal moment generating function for X, we set t₂ to zero in the joint moment generating function, i.e., Mx(t) = Mx,y(t, 0). Similarly, to find the marginal moment generating function for Y, we set t₁ to zero, i.e., My(t) = Mx,y(0, t).

Let's consider X first. Plugging in t₂ = 0 into the joint moment generating function, we get Mx(t) = E[eᵗˣ]. This is simply the moment generating function of X alone.

Now let's find the marginal moment generating function for Y. By setting t₁ = 0 in the joint moment generating function, we get My(t) = E[e²ʸ]. Again, this is the moment generating function for Y alone.