Question

"Given the moment generating function M_S(t) = (1/6) * (e^t + e^(2t) + e^(3t) + e^(4t) + e^(5t) + e^(6t) for the random variable S, calculate the standard deviation for S, denoted as σ(S) = Var[S]."

Answer 1

To find the standard deviation of a discrete random variable with a given MGF, we calculate the first and second derivatives to find the expected value and the second moment. These values are then used to compute the variance, and the standard deviation is the square root of the variance.

Step-by-step explanation:

To calculate the standard deviation (σ) of a discrete random variable using the moment generating function (MGF), we first need to find the expected value (μ) and the second moment (E[S2]), which will then allow us to find the variance (σ2). The variance is the expectation of the squared deviation of a random variable from its mean. The standard deviation is the square root of the variance.

For the given MGF M_S(t) = (1/6) * (et + e2t + e3t + e4t + e5t + e6t), we start by finding the first and second derivatives needed to calculate the moments:

1. M_S'(t) = d/dt M_S(t) = (1/6) * (et + 2e2t + 3e3t + 4e4t + 5e5t + 6e6t)
2. M_S''(t) = d2/dt2 M_S(t) = (1/6) * (et + 4e2t + 9e3t + 16e4t + 25e5t + 36e6t)

Then we find the first and second moments at t = 0:

• μ = E[S] = M_S'(0) = (1/6)(1 + 2 + 3 + 4 + 5 + 6) = 3.5
• E[S2] = M_S''(0) = (1/6)(1 + 4 + 9 + 16 + 25 + 36) = 15.1666...

Now, we compute the variance:

σ2 = E[S2] - (μ)2 = 15.1666... - (3.5)2 = 2.9166...

Finally, the standard deviation σ(S) is the square root of the variance: σ(S) = √2.9166... = 1.7078...