Linear Combinations of Independent Random Variables
Variance and Standard Deviation
This Exam P sample reference tests Exponential Distribution. This is a variance calculation for a weighted sum of two independent exponential durations. Equating the stated total variance to 25+16c² gives c=0.357071, so choice C is correct.
How to solve this Exponential Distribution question
Setup
Setup
Let U and V denote the two durations. An exponential variable's variance is the square of its mean.
Var(U)=52=25,Var(V)=42=16
Model
Model
The total monetary loss is a linear combination of the durations. Independence removes the covariance term, while each rate is squared in the variance.
L=U+cV
Var(L)=Var(U)+c2Var(V)=25+16c2
Compute
Compute
Square the supplied total standard deviation and solve the resulting equation for the nonnegative rate.
5.202=25+16c2
c2=1627.04−25=0.1275
c=0.1275=0.3570714214
Answer
Answer
The nearest listed rate is 0.36.
c≈0.36(C)
Test the method on unseen problems
Probability Multivariate & Risk Models Practice Workbook
30 original multivariate random variables problems with complete solutions and wrong-choice analysis. Written for Exam P, not copied from the SOA set. One-time $19 PDF.
These notes identify the calculation error associated with each wrong letter when that error is reproducible.
AThis adds transformed standard deviations linearly, 5+4c=5.20, which gives c=0.05 and incorrectly treats independent components as perfectly positively dependent.
BThis stops at c²=(27.04-25)/16=0.1275 and reports the squared rate, rounded to 0.13, instead of taking its positive square root.
DThis treats the exponential means as variances, solving 5+4c²=27.04 and obtaining c=√(5.51)=2.35.
EThis both substitutes means for variances and omits the final square root, reporting (27.04-5)/4=5.51.
Original practice · fully worked
Original variant: variability of a laboratory timing index
An automated laboratory records independent stage times U and V. The times are exponential with rates 0.25 and 0.50 per minute, respectively. A diagnostic index is R=2U-3V. Calculate the standard deviation of R.
A 2.000
B 3.606
C 6.000
D 10.000
E 14.000
Answer and calculation
The exponential variances are 16 and 4. Independence gives Var(R)=2²(16)+(-3)²(4)=100, so the standard deviation is 10 and choice D is correct.
Setup
Setup
Convert each exponential rate to a variance.
Var(U)=0.2521=16,Var(V)=0.5021=4
Model
Model
Square both coefficients and add the independent variance contributions.
Var(R)=22Var(U)+(−3)2Var(V)
Compute
Compute
Evaluate the variance and then take its square root.
