Engineering Optimization: Theory and Practice, Fourth Edition

664 Stochastic Programming

distribution,

fV(ν)=







1

V 0

e−(ν/V^0 ) forν≥ 0

0 forν < 0

whereV 0 is the mean velocity, derive the density function for the head lossH.

11.7 The joint density function of two random variablesXandYis given by

fX,Y(x, y)=

{

3 x^2 y+ 3 y^2 x for 0≤x≤ 1 , 0 ≤y≤ 1

0 elsewhere

Find the marginal density functions ofXandY.

11.8 Steel rods, manufactured with a nominal diameter of 3 cm, are considered acceptable

if the diameter falls within the limits of 2.99 and 3.01 cm. It is observed that about 5

% are rejected oversize and 5 % are rejected undersize. Assuming that the diameters

are normally distributed, find the standard deviation of the distribution. Compute the

proportion of rejects if the permissible limits are changed to 2.985 and 3.015 cm.

11.9 Determine whether the random variablesXandYare dependent or independent when

their joint density function is given by

fX,Y(x, y)=

{

4 xy for 0≤x≤ 1 , 0 ≤y≤ 1

0 elsewhere

11.10 Determine whether the random variablesXandYare dependent or independent when

their joint density function is given by

fX,Y(x, y)=







1

4 π^2

[1−sin(x+y)] for−π≤x≤π, −π≤y≤π

0 elsewhere

11.11 The stress level at which steel yields (X)has been found to follow normal distribution.

For a particular batch of steel, the mean and standard deviation ofXare found to be

4000 and 300 kgf/cm^2 , respectively. Find

(a)The probability that a steel bar taken from this batch will have a yield stress between

3000 and 5000 kgf/cm^2

(b)The probability that the yield stress will exceed 4500 kgf/cm^2

(c)The value ofXat which the distribution function has a value of 0.10

11.12 An automobile body is assembled using a large number of spot welds. The number of

defective welds (X)closely follows the distribution

P (X=d)=

e−^22 d

d!

, d= 0 , 1 , 2 ,...

Find the probability that the number of defective welds is less than or equal to 2.