P1.T2.303 Mean and variance of continuous probability density functions (pdf)
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)?
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?
303.3. Assume the following probability density function (pdf) for a random variable X:
What is the variance of X?