# Question

If a particle is dropped at time t = 0, physical theory indicates that the relationship between the distance traveled r and the time elapsed t is r = gtk for some positive constants g and k. A transformation to linearity can be obtained by taking logarithms: log r = log g + k log t.

By letting y = log r, A = log g, and x = log t, this relation becomes y = A + kx. Due to random error in measurement, however, it can be stated only that E(Y׀x) = A + kx. Assume that Y is normally distributed with mean A + kx and variance σ2.

A physicist who wishes to estimate k and g performs the following experiment: At time 0 the particle is dropped. At time t the distance r is measured. He performs this experiment five times, obtaining the following data (where all logarithms are to base 10).

(a) Obtain least-squares estimates for k and log g, and forecast the distance traveled when log t = +3.0.

(b) Starting with a forecast for log r when log t = 0, use the exponential smoothing method with an initial estimate of log r = –3.95 and α = 0.1, that is,

Forecast of log r (when log t = 0) = 0.1(–2.12) + 0.9(3.95), to forecast each log r for all integer log t through log t = +3.0.

(c) Repeat part (b), except adjust the exponential smoothing method to incorporate a trend factor into the underlying model as described in Sec. 27.6. Use an initial estimate of trend equal to the slope found in part (a). Let β = 0.1.

By letting y = log r, A = log g, and x = log t, this relation becomes y = A + kx. Due to random error in measurement, however, it can be stated only that E(Y׀x) = A + kx. Assume that Y is normally distributed with mean A + kx and variance σ2.

A physicist who wishes to estimate k and g performs the following experiment: At time 0 the particle is dropped. At time t the distance r is measured. He performs this experiment five times, obtaining the following data (where all logarithms are to base 10).

(a) Obtain least-squares estimates for k and log g, and forecast the distance traveled when log t = +3.0.

(b) Starting with a forecast for log r when log t = 0, use the exponential smoothing method with an initial estimate of log r = –3.95 and α = 0.1, that is,

Forecast of log r (when log t = 0) = 0.1(–2.12) + 0.9(3.95), to forecast each log r for all integer log t through log t = +3.0.

(c) Repeat part (b), except adjust the exponential smoothing method to incorporate a trend factor into the underlying model as described in Sec. 27.6. Use an initial estimate of trend equal to the slope found in part (a). Let β = 0.1.

## Answer to relevant Questions

Suppose that the relation between Y and x is given by E(Y׀x) = Bx, Where Y is assumed to be normally distributed with mean Bx and known variance σ2. Also n independent pairs of observations are taken and are denoted by x1, ...A company uses exponential smoothing with α = ½ to forecast demand for a product. For each month, the company keeps a record of the forecast demand (made at the end of the preceding month) and the actual demand. Some of ...Refer to the financial risk analysis example presented in Sec. 28.4, including its results shown in Fig. 28.15. Think-Big management is quite concerned about the risk profile for the proposal. Two statistics are causing ...The leading brewery on the West Coast (labeled A) has hired an OR analyst to analyze its market position. It is particularly concerned about its major competitor (labeled B). The analyst believes that brand switching can be ...Consider the inventory example presented in Sec. 29.1 except that demand now has the following probability distribution: P{D = 0} = 1/4, P{D = 2} = 1/4, P{D = 1} = 1/2, P{D > 3} = 0. The ordering policy now is changed to ...Post your question

0