To determine the expected value E[Y] of the discrete random variable Y, we can use the property of the Moment Generating Function (MGF) that M'_{Y}(0)=E[Y]. That means that the first derivative of the MGF evaluated at t=0 gives us the expected value.
We are given the Moment Generating Function (MGF) of Y, which is M_{Y}(t) = 1/7 * e^(2t) + 3/7 * e^(3t) + 2/7 * e^(5t) + 1/7 * e^(8t).
To find E[Y], we first need to calculate the derivative of M_{Y}(t) with respect to t.
The derivative of e^(at) with respect to t is a * e^(at), so differentiating M_{Y}(t) gives us :
M'_{Y}(t) = 1/7 * 2 * e^(2t) + 3/7 * 3 * e^(3t) + 2/7 * 5 * e^(5t) + 1/7 * 8 * e^(8t).
Now, to find E[Y], we evaluate M'_{Y}(t) at t=0:
E[Y] = M'_{Y}(0) = 1/7 * 2 + 3/7 * 3 +2/7 * 5 +1/7 * 8,
which simplifies to approximately 4.14285714285714.
Therefore, the expected value of the discrete random variable Y is approximately 4.14285714285714.