082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

944 Chapter 29 Statistics and probability distributions
Each termisof the form (value) ×(probability). Thus,
∑i=5
expected value = x i
P(x i
)
i=1
The symbol, µ, isused todenote the expected value of a random variable.
If a discrete random variable can take values
x 1
,x 2
,...,x n
with probabilitiesP(x 1
),P(x 2
),...,P(x n
), then
∑i=n
expected value ofx = µ = x i
P(x i
)
i=1
Example29.8 Arandom variable,y, has a known probability distributiongiven by
y 2 4 6 8 10
P(y) 0.17 0.23 0.2 0.3 0.1
Find the expected value ofy.
Solution We have
µ = expectedvalue = 2(0.17) +4(0.23) +6(0.2) +8(0.3) +10(0.1) = 5.86
29.7.2 Expectedvalueofacontinuousrandomvariable
Suppose a continuous random variable,x, has p.d.f. f (x),a x b. The probability
thatxliesinavery smallinterval, [x,x + δx], is
∫ x+δx
x
f(t)dt
Since the interval is very small, f will vary only slightly across the interval. Hence the
probability is approximately f (x)δx: see Figure 29.6. The contribution to the expected
value as a resultofthis interval is
(value) ×(probability)
that is,xf(x)δx. Summing all such termsyields
expected value = µ =
∫ b
a
xf(x)dx