1 pointTwo fair dice are rolled simultaneously. Let 𝑌Y represent the sum of the numbers appearing on the upper faces of the dice. Determine the moment-generating function (MGF) of 𝑌−𝐸[𝑌]Y−E[Y]. Select all correct options from below.𝑒−3.5𝑡⋅16(𝑒𝑡+𝑒2𝑡+𝑒3𝑡+𝑒4𝑡+𝑒5𝑡+𝑒6𝑡)e −3.5t ⋅ 61​ (e t +e 2t +e 3t +e 4t +e 5t +e 6t )16(𝑒𝑡+𝑒2𝑡+𝑒3𝑡+𝑒4𝑡+𝑒5𝑡+𝑒6𝑡)61​ (e t +e 2t +e 3t +e 4t +e 5t +e 6t )[𝑒−3.5𝑡⋅16(𝑒𝑡+𝑒2𝑡+𝑒3𝑡+𝑒4𝑡+𝑒5𝑡+𝑒6𝑡)]2[e −3.5t ⋅ 61​ (e t +e 2t +e 3t +e 4t +e 5t +e 6t )] 2 𝑒−7𝑡⋅136(𝑒−5𝑡+2𝑒−4𝑡+3𝑒−3𝑡+4𝑒−2𝑡+5𝑒−𝑡+6+5𝑒𝑡+4𝑒2𝑡+3𝑒3𝑡+2𝑒4𝑡+𝑒5𝑡)e −7t ⋅ 361​ (e −5t +2e −4t +3e −3t +4e −2t +5e −t +6+5e t +4e 2t +3e 3t +2e 4t +e 5t )

The moment-generating function (MGF) of a random variable is a function that generates the moments of the distribution. The MGF of a random variable Y is given by M(t) = E[e^(tY)].

In this case, Y is the sum of the numbers appearing on the upper faces of two fair dice, and we are asked to find the MGF of Y - E[Y].

First, we need to find the expected value E[Y]. Since each die has 6 faces, each with equal probability of 1/6, the expected value E[Y] of the sum of two dice is E[Y] = 2 * (1/6 * (1 + 2 + 3 + 4 + 5 + 6)) = 7.

So, we are asked to find the MGF of Y - 7.

The MGF of Y - 7 is given by M(t) = E[e^(t(Y - 7))].

This simplifies to M(t) = e^(-7t) * E[e^(tY)].

The MGF of the sum of two dice Y is given by E[e^(tY)] = (1/36) * Σ e^(tx), where the sum is over all possible values of x (the sum of two dice), x = 2, 3, ..., 12.

So, the MGF of Y - 7 is M(t) = e^(-7t) * (1/36) * (e^(2t) + e^(3t) + 2e^(4t) + 2e^(5t) + 3e^(6t) + 3e^(7t) + 4e^(8t) + 4e^(9t) + 3e^(10t) + 3e^(11t) + 2e^(12t) + 2e^(13t) + e^(14t) + e^(15t)).

So, the correct option is 𝑒−7𝑡⋅136(𝑒−5𝑡+2𝑒−4𝑡+3𝑒−3𝑡+4𝑒−2𝑡+5𝑒−𝑡+6+5𝑒𝑡+4𝑒2𝑡+3𝑒3𝑡+2𝑒4𝑡+𝑒5𝑡).