Question

Let \(X\) be a number selected at random from a set of numbers \(\\{51,52, \ldots, 100\\}\). Approximate \(E(1 / X)\) Hint: Find reasonable upper and lower bounds by finding integrals bounding \(E(1 / X)\).

Short answer

The approximated expected value E[1 / X] for X in the set {51,52,...,100} is between approximately 0.01922 and 0.01996.

Step-by-step solution

Identify the Distribution and the Function

The variable \(X\) is chosen randomly from the set {51, 52,...,100}, which implies a uniform distribution over this range. The function of \(X\) in question is \(1 / X\). So we are trying to find \(E[1 / X]\) over the range of 51 to 100.

Set Up the Bounding Integrals

The hint suggests finding integrals that 'bound' \(E[(1 / X)]\). This term typically refers to finding upper and lower bounds for the expected value. Since \(1 / X\) is a decreasing function in the range of \(X\) in question (51 to 100), it will follow that \(1 / 51 <= 1 / X <= 1 / 100\). This gives us the bounds for \(1 / X\), which we can now integrate.

Evaluate the Integrals

The integral bounds are from 51 to 100, and the distribution is uniform, so each case should be divided by the size of the set, which is 100 - 51 + 1 = 50. The upper bound integral will therefore be \( \int_{51}^{100} 1 / x dx / 50 = [ln|x|]_{51}^{100} / 50 = (ln|100| - ln|51|) / 50 \approx 0.01996 \). The lower bound integral will be \( \int_{51}^{100} 1 / 51 dx / 50 = (1 / 51) * [x]_{51}^{100} / 50 = 49 / 2550 = 0.01922 \).

Conclude the Solution

Thus you've obtained two bounds for \(E[1 / X]\), which will be between these two numbers. The exact value can't be found because the distribution specified is discrete whereas the bounding integrals are found using continuous integral calculus. However, the bounds provide a reasonable estimate.