Question

A contractor has submitted bids on three state jobs: an office building, a theater, and a parking garage. State rules do not allow a contractor to be offered more than one of these jobs. If this contractor is awarded any of these jobs, the profits earned from these contracts are $10 million from the office building, $5 million from the theater, and $2 million from the parking garage. His profit is zero if he gets no contract. The contractor estimates that the probabilities of getting the office building contract, the theater contract, the parking garage contract, or nothing are .15, .30, .45, and .10, respectively.

1. Write the probability distribution of x .
2. Find the mean and standard deviation of x .

Answer 1

The probability distribution of x is

Profits in millions (x) Probability of getting contract (P(x))

$10 0.15

$ 5 0.30

$ 2 0.45

$ 0 0.10

The mean is
`E(X) = 3.9 millions`

The standard deviation is
`s = 3.01 \  millons`

Explanation:

Generally the probability distribution of x is

Profits in millions (x) Probability of getting contract (P(x))

$10 0.15

$ 5 0.30

$ 2 0.45

$ 0 0.10

Generally the mean is mathematically represented as

`E(X) =  x  *  p(x)`

=>
`E(X) =  [10 *  0.15 ]  +  [5 *0.30 ] + [2 *  0.45 ] + [0 * 0.10]`

=>
`E(X) = 3.9 millions`

Now

`E(X^2) =  x^2 * p(x)`

=>
`E(X^2) =  (10 ^2 *   0.15) +  (5  ^2 *   0.30) +(2  ^2 *   0.45) + (0 * 0.10)`

=>
`E(X^2) = 24.3`

Gnerally the variance is mathematically represented as

`V(X) =  E(X^2) - [E(X)]^2`

=>
`V(X) =  24.3 -  3.9^2`

=>
`V(X) =  9.09`

Gnerally the standard deviation is mathematically represented as

`s =  √(V(X))`

=>
`s =  \sqrt{9.09`

=>
`s = 3.01`