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A bank accepts rolls of pennies and gives 50 cents credit to a customer without counting the contents. Assume that a roll contains 49 pennies 30 percent of the time, 50 pennies 60 percent of the time, and 51 pennies 10 percent of the time. (a) Find the expected value and the variance for the amount that the bank loses on a typical roll.

Respuesta :

Answer:

E[X] = 0.2 and Var(X) = 0.36

Step-by-step explanation:

Let X be the money in cents that the bank loses on one roll.

Then P(X = 1) = 0.3, P(X = 0) = 0.6,  and P(X = −1) = 0.1.

That yields an expected loss for the bank of E[X] = 0.3 + 0 - 0.1 = 0.2

As can be seen from the question, the bank will be in loss only in the case when they get 49 pennies 30 percent of time and when they will get 50 pennies, it means they do not have any loss. Finally, bank will get the benefit when they get 51 pennies, it mean no loss but profit.

That is why to calculate Expected values we used P =0.3 for loss and, p=0 for 50 pennies and -0.1 for 51 pennies.

In the end we get E[X] = 0.2

and the variance is (1-E[X])^2 * P(X)

Variance[X] = (1 − 0.2)^2  * 0.3 + (0 − 0.2)^2 *0.6 + (−1 − 0.2)^2 * 0.1

Var[X] = 0.192+0.024+0.144

Var[X] = 0.36

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