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P1.T2.303 Mean and variance of continuous probability density functions (pdf)

Suzanne Evans

Well-Known Member
AIMs: Define, calculate, and interpret the mean, standard deviation, and variance of a random variable.


303.1. Assume a continuous probability density function (pdf) is given by f(x) =a*x such that 0 ≤ x ≤ 12, where a is a constant (we can retrieve this constant, knowing this is a probability density function):

What is the mean of (x)?

a. 5.5
b. 6.0
c. 8.0
d. 9.3

303.2. Assume a continuous probability density function (pdf) be given by f(x) = a*x^2 such that 0 ≤ x ≤ 3, where a is a constant (that we can find).

Let us arbitrarily define the unexpected loss (UL) as the difference between this distribution's mean and its 5.0% quantile function; i.e., UL(X) = mean (X) - inverse CDF(5%)(X). We could call this a 95% relative VaR since it is relative to the mean. What is this UL?

a. 0.62
b. 1.14
c. 2.05
d. 3.37

303.3. Assume the following probability density function (pdf) for a random variable X:
What is the variance of X?

a. 2.0
b. 3.3
c. 4.1
d. 5.7

Hi Suzanne - could you please be so kind to post answers and formula to solve these two last questions 303.2 and 303.3. Of course - if it is not prohibited. Thank you very much beforehand! I was trying to solve these two and used integral but unavailingly...

David Harper CFA FRM

David Harper CFA FRM
Staff member
Hi Monte Carlo - We totally appreciate your interest, but you've got to understand: the three questions above took me about 1.5 hours to write. New question sets (of three) require of me daily 1.5 up to 3.5 hours per day, by the time i calibrate them and cross-reference check etc ... this sausage is all hand-made with tender loving care ;) .... so we need to really do rely on customers to pay us for the service. Thanks for your understanding,
Yes of course I understand that, I`m not yet paid for the service because I haven`t yet registered for the FRM - so it just a matter of time... Thanks anyway:) I fully understand that it`s hard to write these questions and then to check everything...