Question

A heavy-equipment salesperson can contact either one or two customers per day with probability 1/3 and 2/3, respectively. each contact will result in either no sale or a $50,000 sale, with the probabilities 0.9 and 0.1, respectively. find the mean and standard deviation of the daily sales. (hint: first obtain the probability distribution for daily sales.)

Answer 1

Answer: The mean of daily sales is $8,333.33 and the standard deviation of daily sales $19,507.83.

We arrive at the answer as follows:

1. Obtaining probability distribution for daily sales

On any day, the number of sales may be either 0 ,1 or 2.

We compute each of the probabilities as:

$P(X=0) = P(1 customer and no sales) + P( 2 customers and no sales)$

$P(X=0) = (\frac{1}{3}*0.9) + (\frac{2}{3}* [0.9*0.9])$

$\mathbf{P(X=0) = 0.84}$

$P(X=1) = P(1 customer and 1 sale) + P( 1st customer and sales + 2nd customer and no sales) + P( 1 customers and no sales + 2nd customers and sales)$

$P(X=1) = (\frac{1}{3}*0.1) + (\frac{2}{3}*\frac{1}{10}*\frac{9}{10}) + (\frac{2}{3}*\frac{9}{10}*\frac{1}{10})$

$\mathbf{P(X=1) = 0.153333333}$

Since we know the P(X=0) and P(X=1), we can calculate P(X=2) as:

$P(X=2) = 1 - P(X=0)- P(X=1)$

$P(X=2) = 1 - 0.84 - 0.153333333$

$\mathbf{P(X=2) = 0.006666667}$

Now that we have the Probability distribution of daily sales, we can compute the mean of daily sales as follows:

Let V be value of sales per day, the mean is

Daily Sales       P           Value of sales -v                       v*P

     0           0.8400                     0                                        0

    1             0.1533               50000                           7666.666667

    2              0.0067              100000                              666.666667

Total               1                                                             8333.333333

The total in the table above is the mean of daily sales.

We calculate Standard Deviation as

$\sigma = \sqrt{[(0 - 8333.3333)^{2})*0.84] + [(7666.666667 - 8333.3333)^{2})*0.1533]+[(666.666667 - 8333.3333)^{2})* 0.0067]}$

$\mathbf{\sigma = 19507.83318}$