11.1 Mathematical expectation; general random variables  (Page 6/7)

Stocking for random demand (see Exercise 4 From "problems on functions of random variables")

The manager of a department store is planning for the holiday season. A certain item costs c dollars per unit and sells for p dollars per unit. If the demand exceeds the amount m ordered, additional units can be special ordered for s dollars per unit ( $s>c$ ). If demand is less than amount ordered, the remaining stock can be returned (or otherwise disposed of) at r dollars per unit ( $r<c$ ). Demand D for the season is assumed to be a random variable with Poisson $\left(\mu \right)$ distribution. Suppose $\mu =50,c=30,p=50,s=40,r=20$ . What amount m should the manager order to maximize the expected profit?

PROBLEM FORMULATION

Suppose D is the demand and X is the profit. Then

• For $D\le m,X=D(p-c)-(m-D)(c-r)=D(p-r)+m(r-c)$
• For $D>m,X=m(p-c)+(D-m)(p-s)=D(p-s)+m(s-c)$

It is convenient to write the expression for X in terms of I _M , where $M=(-\infty ,m]$ . Thus

Then $E\left[X\right]=(p-s)E\left[D\right]+m(s-c)+(s-r)E\left[I_M\left(D\right)D\right]-(s-r)mE\left[I_M\left(D\right)\right]$ .

ANALYTIC SOLUTION

For $D\sim$ Poisson $\left(\mu \right)$ , $E\left[D\right]=\mu \text{and}E\left[I_M\left(D\right)\right]=P(D\le m)$

Hence,

Because of the discrete nature of the problem, we cannot solve for the optimum m by ordinary calculus. We may solve for various m about $m=\mu$ and determine the optimum. We do so with the aid of MATLAB and the m-function cpoisson.

A direct, solution may be obtained by MATLAB, using finite approximation for the Poisson distribution.

APPROXIMATION

An advantage of the second solution, based on simple approximation to D , is that the distribution of gain for each m could be studied — e.g., the maximum and minimum gains.

— $\square$

A jointly distributed pair

Suppose the pair $\{X,Y\}$ has joint density $f_{XY}(t,u)=3u$ on the triangular region bounded by $u=0$ , $u=1+t$ , $u=1-t$ (see [link] ). Let $Z=g(X,Y)=X^2+2XY$ . Determine $E\left[Z\right]$ .

Analytic solution

${\displaystyle E\left[Z\right]=\int \int (t^2+2tu)f_{XY}(t,u)dudt}$

$$=3{\int }_{-1}^0{\int }_0^{1+t}(t^{2}u+2t{u}^2)dudt+3{\int }_0^1{\int }_0^{1-t}(t^{2}u+2t{u}^2)dudt=1/10$$

APPROXIMATION

tuappr Enter matrix [a b]of X-range endpoints [-1 1] Enter matrix [c d]of Y-range endpoints [0 1] Enter number of X approximation points 400Enter number of Y approximation points 200 Enter expression for joint density 3*u.*(u<=min(1+t,1-t)) Use array operations on X, Y, PX, PY, t, u, and PG = t.^2 + 2*t.*u; % g(X,Y) = X^2 + 2XY EG = total(G.*P) % E[g(X,Y)]EG = 0.1006 % Theoretical value = 1/10 [Z,PZ]= csort(G,P); % Distribution for Z EZ = Z*PZ' % E[Z]from distribution EZ = 0.1006 Got questions? Get instant answers now!