The manager of a stockroom in a factory knows from his study

The manager of a stockroom in a factory knows from his study of

records that the daily demand (number of times used) for a certain tool has the

following probability distribution: no uses, 0.1; 1 use, 0.5; and 2 uses, 0.4. For

example, 50% of the daily records show that the tool was used one time.

Letting X denote the daily demand, find the following:

(i) E(X) and V(X).

(ii) If it costs the factory $10 each time the tool is used, find the mean and the

variance of the daily costs of using this tool.

Solution

Consider the table:          
          
x   P(x)   x P(x)   x^2 P(x)
0   0.1   0   0
1   0.5   0.5   0.5
2   0.4   0.8   1.6
          
Totals       1.3   2.1
       =E(x)   =E(x^2)
          
Thus,          
          
i)

E(x) =    1.3      
Var(x) = E(x^2) - E(x)^2 =    0.41

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ii)

E(10x) = 10E(x) = 10*1.3 = $13 [mean cost]
Var(10x) = 10^2 Var(x) = 100*0.41 = 41 [answer, variance, daily cost]