2.4 – A Simple PDF Example

Solution

Part 1)

Therefore, k = 1/10.

Notice that f(x) ≥ 0 for all x. Also notice that we can rewrite this PDF in the obvious way so that it is defined for all real numbers:

Part 2)

Using our value for k from Part 1:

Therefore, Pr(1 ≤ X ≤ 2) is 4/5.

Part 3)

Using the Fundamental Theorem of Calculus, the CDF of X at x in [0,2] is

We can also easily verify that F(x) = 0 for all x < 0 and that F(x) = 1 for all x > 2.

Part 4)

Since X is a continuous random variable, we immediately know that the probability that it equals any one particular value must be zero. More directly, we compute