308
HL Unit 9 (Probability) by 1 x 8 27 f (x) = 8x2 0,
for 0 ≤ x ≤ 3, for 3 < x ≤ a, otherwise.
Ans:
Example 9.5.3 The basic shape of the bell curve is given by e−x R∞ 2 (a) Find 0 e−x /2 dx R∞ 2 (b) Find −∞ e−x /2 dx (c) Let f (x) = ke−x
2
2
/2
54 11
.
/2
. Find a value of k that will make h f (x) bei a probability density function. Ans: √12π
Expected Value Definition 9.15. The expected value E (X), of a continuous random variable X with probability density function f (x), is given by Z ∞ E (X) = x f (x) dx −∞
Example 9.5.4 Suppose that X is the lifetime of a Powermate bat- 3 tery, in months, and that the pdf is given by f (x) = 32 4x − x2 for 0 < x < 4, and f (x) = 0, elsewhere. Calculate E (() X), the mean of X.
[Ans: two months]
Example 9.5.5 (MM 5/99) A continuous random variable X has the probability density function 2x , for 0 ≤ x ≤ 5; f (x) = 25 0, elsewhere. Find the value of E (X). Mr. Budd, compiled September 29, 2010