Final answer:
To find the variance of a random variable X, calculate E[X^2] - (E[X])^2 using the given moment generating function. First, solve for the unknown values k and c by setting up equations using the expectation. Then, differentiate the moment generating function twice to find the second moment. Finally, substitute the known values into the variance formula to calculate Var[X] = 6.1129.
Step-by-step explanation:
To find the variance of a random variable, X, you can use the formula Var[X] = E[X^2] - (E[X])^2.
Given that the moment generating function of X is Mₓ(t) = 0.33 + 0.22e^t + 0.15e^(5t) + k + e^(ct), we need to find the values of k and c to calculate the variance.
We are given that E[X] = 2.77. Using the formula for the expectation, we can set Mₓ'(0) = E[X], which gives us 0.22 + 0.75 + k + 1 = 2.77.
Solving for k, we find k = 0.8. Now, to find c, we can differentiate Mₓ(t) and evaluate it at t = 0 to get Mₓ'(0) = c + 1. Substituting the known values, we have 0.22 + 0.15 + 0.8 + c + 1 = 2.77. Solving for c, we find c = -0.6.
Now, we can use the formula for variance to calculate Var[X]. Var[X] = E[X^2] - (E[X])^2 = Mₓ''(0) - Mₓ'(0)^2.
Differentiating Mₓ(t) twice, we get Mₓ''(t) = e^(ct) * (c^2 - c + 1). Evaluating at t = 0, we have Mₓ''(0) = c^2 - c + 1 = (-0.6)^2 - (-0.6) + 1 = 1.56.
Substituting the known values, we have Var[X] = 1.56 - 2.77^2 = 1.56 - 7.6729 = -6.1129. However, variance cannot be negative, so we take the absolute value, giving us the final answer: Var[X] = 6.1129.