Variance (Page 3/4)

Applied probability Page 3 / 4

$Z = g (X, Y)$ ( Example 10 From "mathematical expectation: simple random variables")

We use the same joint distribution as for Example 10 from "Mathematical Expectation: Simple Random Variables" and let $g (t, u) = t^{2} + 2 t u - 3 u$ . To set up for calculations, we use jcalc.

jdemo1                      % Call for data
jcalc                       % Set upEnter JOINT PROBABILITIES (as on the plane)  P
Enter row matrix of VALUES of X  XEnter row matrix of VALUES of Y  Y
 Use array operations on matrices X, Y, PX, PY, t, u, and PG = t.^2 + 2*t.*u - 3*u;    % Calculation of matrix of [g(t_i, u_j)]
EG = total(G.*P)            % Direct calculation of E[g(X,Y)]EG =   3.2529
VG = total(G.^2.*P) - EG^2  % Direct calculation of Var[g(X,Y)]VG =  80.2133
[Z,PZ]= csort(G,P);        % Determination of distribution for Z
EZ = Z*PZ'                  % E[Z]from distribution
EZ =   3.2529VZ = (Z.^2)*PZ' - EZ^2      % Var[Z] from distributionVZ =  80.2133

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A function with compound definition ( Example 12 From "mathematical expectation; general random variables")

Suppose $X \sim$ exponential (0.3). Let

Z = \{\begin{matrix} X^{2} & for X \leq 4 \\ 16 & for X > 4 \end{matrix}) = I_{[0, 4]} (X) X^{2} + I_{(4, \infty]} (X) 16

Determine $E [Z]$ and $Var [Z]$ .

ANALYTIC SOLUTION

E [g (X)] = \int g (t) f_{X} (t) d t = \int_{0}^{\infty} I_{[0, 4]} (t) t^{2} 0.3 e^{- 0.3 t} d t + 16 E [I_{(4, \infty]} (X)]

= \int_{0}^{4} t^{2} 0.3 e^{- 0.3 t} d t + 16 P (X > 4) \approx 7.4972 (by Maple)

Z^{2} = I_{[0, 4]} (X) X^{4} + I_{(4, \infty]} (X) 256

E [Z^{2}] = \int_{0}^{\infty} I_{[0, 4]} (t) t^{4} 0.3 e^{- 0.3 t} d t + 256 E [I_{(4, \infty]} (X)] = \int_{0}^{4} t^{4} 0.3 e^{- 0.3 t} d t + 256 e^{- 1.2} \approx 100.0562

Var [Z] = E [Z^{2}] - E^{2} [Z] \approx 43.8486 (by Maple)

APPROXIMATION

To obtain a simple aproximation, we must approximate by a bounded random variable. Since $P (X > 50) = e^{- 15} \approx 3 \cdot 10^{- 7}$ we may safely truncate X at 50.

tappr
Enter matrix [a b]of x-range endpoints  [0 50]
Enter number of x approximation points  1000Enter density as a function of t  0.3*exp(-0.3*t)
Use row matrices X and PX as in the simple caseM = X<= 4;
G = M.*X.^2 + 16*(1 - M);  % g(X)EG = G*PX'                 % E[g(X)]
EG =  7.4972VG = (G.^2)*PX' - EG^2     % Var[g(X)]
VG = 43.8472               % Theoretical = 43.8486[Z,PZ] = csort(G,PX);      % Distribution for Z = g(X)EZ = Z*PZ'                 % E[Z] from distributionEZ =  7.4972
VZ = (Z.^2)*PZ' - EZ^2     % Var[Z]VZ = 43.8472

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Stocking for random demand ( Example 13 From "mathematical expectation; general 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 the 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 $(μ)$ distribution. Suppose $μ = 50, c = 30, p = 50, s = 40, r = 20$ . 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 $D \leq m, X = D (p - c) - (m - D) (c - r) = D (p - r) + m (r - c)$
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

X = I_{M} (D) [D (p - r) + m (r - c)] + [1 - I_{M} (D)] [D (p - s) + m (s - c)]

= D (p - s) + m (s - c) + I_{M} (D) [D (p - r) + m (r - c) - D (p - s) - m (s - c)]

= D (p - s) + m (s - c) + I_{M} (D) (s - r) [D - m]

Then

E [X] = (p - s) E [D] + m (s - c) + (s - r) E [I_{M} (D) D] - (s - r) m E [I_{M} (D)]

We use the discrete approximation.

APPROXIMATION

>>mu = 50;>>n = 100;>>t = 0:n;>>pD = ipoisson(mu,t);         % Approximate distribution for D>>c  = 30;>>p  = 50;>>s  = 40;>>r  = 20;>>m  = 45:55;>>for i = 1:length(m)          % Step by step calculation for various m
    M = t<=m(i);
    G(i,:) = (p-s)*t + m(i)*(s-c) + (s-r)*M.*(t - m(i));end>>EG = G*pD';>>VG = (G.^2)*pD' - EG.^2;>>SG =sqrt(VG);>>disp([EG';VG';SG']')1.0e+04 *
    0.0931    1.1561    0.01080.0936    1.3117    0.0115
    0.0939    1.4869    0.01220.0942    1.6799    0.0130
    0.0943    1.8880    0.01370.0944    2.1075    0.0145
    0.0943    2.3343    0.01530.0941    2.5637    0.0160
    0.0938    2.7908    0.01670.0934    3.0112    0.0174
    0.0929    3.2206    0.0179

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Source: OpenStax, Applied probability. OpenStax CNX. Aug 31, 2009 Download for free at http://cnx.org/content/col10708/1.6

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Variance (Page 3/4)

Z = g ( X , Y ) ( Example 10 From "mathematical expectation: simple random variables")

A function with compound definition ( Example 12 From "mathematical expectation; general random variables")

Stocking for random demand ( Example 13 From "mathematical expectation; general random variables")

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

$Z = g (X, Y)$ ( Example 10 From "mathematical expectation: simple random variables")