Suppose you agree to meet a friend(who is generally late) at a specified time. Assume that you arrive on time and let the random variable T be the length off time you must wait for your friend. If the density function for T is assume to be
f(t)=1/t^2, t>=1(minute)
Find the time you should expect to wait.
To find the expected time you should wait, you need to calculate the expected value or the mean of the random variable T. The expected value, denoted as E(T), is given by the integral of t multiplied by the density function f(t) over its entire range.
In this case, since the density function f(t) is defined as f(t) = 1/t^2 for t >= 1, the range of integration will be from 1 to infinity.
Therefore, the expected value E(T) can be calculated as:
E(T) = ∫(t * f(t)) dt, from 1 to ∞
To solve this integral, we can rewrite f(t) as t^(-2). Thus, the integral becomes:
E(T) = ∫(t * t^(-2)) dt, from 1 to ∞
= ∫(t^(-1)) dt, from 1 to ∞
Integrating t^(-1), we get:
E(T) = [ln(t)] from 1 to ∞
Evaluating the integral limits at infinity and 1:
E(T) = [ln(∞)] - [ln(1)]
E(T) = ∞ - 0
E(T) = ∞
Therefore, the expected time you should wait for your friend is infinite, according to the given density function f(t) = 1/t^2. This suggests that there is no finite expected waiting time.
However, please note that this result is somewhat counterintuitive since it implies an infinite expected waiting time. In practice, it is best to consider reasons behind the assumption of the given density function and its applicability to real-life scenarios.