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Descripción: Solutions Probability and Statistics for Engineers - Solutions Full text book solutiongs for the 8th edition
Statistics for Management and Economics, Eighth Edition Formulas
Numerical Descriptive techniques Population mean N
∑x
µ =
i
i =1
N Sample mean n
x=
∑x
i
i =1
n
Range Largest observation - Smallest observation Population variance N
σ2 =
∑( x − µ )
2
i
i =1
N Sample variance n
s2 =
∑(x i =1
i
− x)2
n −1 Population standard deviation
σ
=
σ2
Sample standard deviation s=
s2
Population covariance
N
σ xy =
∑( x
i
− µ x )( y i − µ y )
i =1
N
Sample covariance n
s xy =
∑( x
i
− x )( y i − y )
i =1
n −1
Population coefficient of correlation
ρ=
σ xy σ xσ y
Sample coefficient of correlation
r=
s xy sxsy
Coefficient of determination R2 = r2 Slope coefficient
b1 =
s xy s x2
y-intercept
b0 = y − b1 x Probability Conditional probability P(A|B) = P(A and B)/P(B) Complement rule P(
A C ) = 1 – P(A)
Multiplication rule P(A and B) = P(A|B)P(B)
Addition rule P(A or B) = P(A) + P(B) - P(A and B) Bayes’ Law Formula
P ( A i | B) =
P( A i ) P ( B | A i ) P( A 1 ) P ( B | A 1 ) + P( A 2 ) P( B | A 2 ) + . . . + P ( A k ) P( B | A k )
Random Variables and Discrete Probability Distributions Expected value (mean)
µ=
E(X) =
∑xP ( x )
all
x
Variance V(x) =
σ2 =
∑( x − µ )
2
all
P( x )
x
Standard deviation σ = σ2
Covariance COV(X, Y) = σxy =
∑( x −µ
Coefficient of Correlation
ρ=
COV ( X , Y )
σxσ y
Laws of expected value 1. E(c) = c 2. E(X + c) = E(X) + c 3. E(cX) = cE(X) Laws of variance 1.V(c) = 0 2. V(X + c) = V(X) 3. V(cX) = c 2 V(X)
x
)( y −µy )P( x , y )
Laws of expected value and variance of the sum of two variables 1. E(X + Y) = E(X) + E(Y) 2. V(X + Y) = V(X) + V(Y) + 2COV(X, Y) Laws of expected value and variance for the sum of more than two variables 1.
2.
k
k
i =1
i =1
k
k
i =1
i =1
E (∑ X i ) = ∑ E ( X i ) V (∑ X i ) = ∑V ( X i ) if the variables are independent
Mean and variance of a portfolio of two stocks E(Rp) = w1E(R1) + w2E(R2) V(Rp) = w12 V(R1) + w22 V(R2) + 2 w1 w 2 COV(R1, R2) = w12 σ12 + w22 σ22 + 2 w1 w 2 ρ σ1 σ2 Mean and variance of a portfolio of k stocks k
E(Rp) =
∑w E ( R ) i
i
i =1 k
V(Rp) =
∑w
2 2 i σi
i =1
k
+2
k
∑∑w w i
j COV
i =1 j =i +1
Binomial probability P(X = x) =
n! p x ( 1 − p ) n −x x! ( n − x )!
µ = np σ2 = np ( 1 − p ) σ = np ( 1 − p )
Poisson probability P(X = x) =
e −µµx x!
( Ri , R j )
Continuous Probability Distributions Standard normal random variable
X −µ
Z =
σ
Exponential distribution µ = σ =1 / λ
P( X > x ) = e −λx P( X < x ) =1 −e −λx
P ( x 1 < X < x 2 ) = P( X < x 2 ) − P( X < x 1 ) = e − λ x1 − e − λ x 2 F distribution
F1−A,ν1 ,ν2 =
1 FA,ν2 ,ν1
Sampling Distributions Expected value of the sample mean E( X ) =µx
=µ
Variance of the sample mean V ( X ) =σ 2x =
σ2 n
Standard error of the sample mean σx =
σ n
Standardizing the sample mean Z =
X −µ σ/
n
Expected value of the sample proportion ˆ ) = µ pˆ = p E (P
Variance of the sample proportion
p (1 − p ) V ( Pˆ ) = σ2pˆ = n Standard error of the sample proportion
p(1 − p) n
σ pˆ =
Standardizing the sample proportion Z =
ˆ −p P p (1 − p ) n
Expected value of the difference between two means E ( X 1 − X 2 ) = µ x1 −x2 = µ1 − µ 2
Variance of the difference between two means
V ( X 1 − X 2 ) = σ x21 − x2 =
σ 12 σ 22 + n1 n2
Standard error of the difference between two means
σ x1 − x2 =
σ12 σ 22 + n1 n 2
Standardizing the difference between two sample means
Z=
( X 1 − X 2 ) − ( µ1 − µ 2 )
σ 12 σ 22 + n1 n2
Introduction to Estimation Confidence interval estimator of µ x ± zα / 2
σ n
Sample size to estimate µ
z σ n = α / 2 W
2
Introduction to Hypothesis Testing Test statistic for µ
z=
x −µ
σ/
n
Inference about One Population Test statistic for µ
t=
x −µ s/
n
Confidence interval estimator of µ s
x ± tα / 2
n
Test statistic for σ2
χ2 =
( n −1 )s 2
σ2
Confidence interval Estimator of σ2 LCL =
UCL =
( n −1 )s 2
χα2 / 2 ( n −1 )s 2
χ12−α / 2
Test statistic for p z=
ˆp − p p( 1 − p ) / n
Confidence interval estimator of p ˆp ± zα / 2
ˆp( 1 − ˆp ) / n
Sample size to estimate p
2
z ˆp( 1 − ˆp ) n = α/ 2 W
Confidence interval estimator of the total of a large finite population s N x ± t α / 2 n
Confidence interval estimator of the total number of successes in a large finite population N ˆp ± zα / 2
ˆp( 1 − ˆp n
Confidence interval estimator of µ when the population is small s
x ± tα / 2
n
N −n N −1
Confidence interval estimator of the total in a small population s N x ± tα / 2 n
N −n N −1
Confidence interval estimator of p when the population is small
p ± zα / 2
ˆp( 1 − ˆp ) n
N −n N −1
Confidence interval estimator of the total number of successes in a small population N ˆp ± z α / 2
ˆp( 1 − ˆp ) n
N −n N −1
Inference About Two Populations Equal-variances t-test of µ1 − µ2
t=
( x1 − x 2 ) − ( µ1 − µ 2 ) 1 1 s 2p + n1 n 2
ν = n1 + n 2 − 2
Equal-variances interval estimator of µ1 − µ2
1 1 ν = n1 + n 2 − 2 ( x1 − x 2 ) ± t α / 2 s 2p + n1 n 2 Unequal-variances t-test of µ1 − µ2
t=
( x1 − x 2 ) − ( µ 1 − µ 2 ) s12 s 22 + n1 n 2