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

946 Chapter 29 Statistics and probability distributions
Solutions
1 1.668
2 3.05
3 1
4 (a) 2.012 (b) 2.68
5 0
6 no
29.8 STANDARDDEVIATIONOFARANDOMVARIABLE
29.8.1 Standarddeviationofadiscreterandomvariable
RecallfromSection29.6thatthestandarddeviationofasetofnumbers, {x 1
,x 2
,...,x n
},
isgiven by
√∑ (xi −x)
standard deviation =
2
n
Nowsupposethatxisadiscreterandomvariablewhichcanhavevaluesx 1
,x 2
,x 3
,...,x n
with respective probabilities of p 1
,p 2
,p 3
,...,p n
; thatis, we have
x x 1 x 2 x 3 ... x n
P(x) p 1 p 2 p 3 ... p n
Let the expected value of x be µ. Then the square of the deviation from the expected
value has anidentical probabilitydistribution:
value (x 1 − µ) 2 (x 2 − µ) 2 ... (x n − µ) 2
probability p 1 p 2 . . . p n
Theexpectedvalueofthemeansquareddeviationisthevariance.Thesymbol σ 2 isused
todenotethe varianceofarandom variable:
variance = σ 2 =
n∑
p i
(x i
− µ) 2
1
As before the standarddeviation isthe squarerootofthe variance:
standard deviation = σ =
√ ∑pi
(x i
− µ) 2
Example29.10 Adiscrete randomvariable has probabilitydistribution
x 1 2 3 4 5
P(x) 0.12 0.15 0.23 0.3 0.2