IAI Actuarial Core Principles · Actuarial Statistics · Generating functions
A random variable X has moment generating function M(t) = (0.3 + 0.7e^t)^10. What are the mean and variance of X?
X is Binomial with n = 10 and p = 0.7, so the mean is 7 and the variance is 10 x 0.7 x 0.3 = 2.1. The other options use the wrong probability or an incorrect variance formula.
- AMean 7, variance 2.1Correct
- BMean 7, variance 4.9
- CMean 3, variance 2.1
- DMean 10, variance 2.1
- Mean 7, variance 0.21
Explanation
The MGF is that of a Binomial(10, 0.7). Mean = np = 7 and variance = npq = 10 x 0.7 x 0.3 = 2.1. Variance 4.9 comes from using p squared times n (wrong formula), and mean 3 uses the failure probability.
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