Question

Suppose continuous random variable X follows a normal distribution with mean 2 and variance 4, i.e. X ~ N(2, 4). What is the mathematical expectation of eˣ? Round your number to 2 digits. Think about why this is larger or smaller than eᴱ[ˣ].

Answer 1

The mathematical expectation of e to the power X, where X is normally distributed with mean 2 and variance 4, is e to the power of 4, which is larger than e to the power of 2, due to Jensen's inequality.

Step-by-step explanation:

The student is asking about the mathematical expectation of e to the power of a normally distributed random variable X, with mean 2 and variance 4 (X ~ N(2, 4)). To find the expectation of eX, known as the moment-generating function of a normal distribution at t=1, we can use the formula E(eX) = e(μ+σ2/2), where μ is the mean and σ2 is the variance of X. Plugging the given values, we calculate E(eX) = e(2+4/2) = e(2+2) = e4.

Now, to understand why the expectation of eX is larger than eE[X], we can refer to Jensen's inequality, which states that for any convex function, such as the exponential function, the expectation of the function applied to a random variable is greater than or equal to the function applied to the expectation of the random variable. Since the exponential function is convex, E(eX) > eE[X]. In our case, E[X] = 2, and thus eE[X] = e2, which is indeed smaller than e4, the expectation of eX.