Question

Imagine a room containing fourteen people, whose ages are as follows: one person aged 14, one person aged 15, three people aged 16, two people aged 22, two people aged 24, five people aged 25 This age distribution is summarized in the following plot, where j is age, and N() is the number of people in the room that have age j (a) Write a formula for the probability that a random person from the room has age j (b) Verify that your probability distribution is normalized, that is to say, the sum of proba- (c) An expectation value is defined as the result of the sum of all options weighted by their We'l call this formula the probability distribution, p(j). bilities for all possible ou probability outcomes is equal to 1 (j)-Σ jp(j) (17) Note that if there are N possible results with equal probability, then pj) and then the expectation value is the same as the mean value. Calculate the expectation value for the age, (). Is this the same as the most probable age? (d) Compute the expectation value for ) (e) Determine the deviations from the expectation value Aj (j- (j) for each age j in this sample. Use these values to calculate the root mean square (r.mA) σ. Note, using our notation, the r.m.s is (18) where the angle brackets in the last equality indicate an expectation value of (Aj) (f) Verify that the following relationship holds in this case: (19)

Answer 1

15P(I6) P(22) + f(2)+ P(25) 4 4- 4- 4- 4- So pti) is no malised + 25 P(25 4e 2 94 52P25 459.54 eats)