Answer:
Step-by-step explanation:
(a)
The bid should be greater than $10,000 to get accepted by the seller. Let bid x be a continuous random variable that is uniformly distributed between
$10,000 and $15,000
The interval of the accepted bidding is ![[ {\rm{\$ 10,000 , \$ 15,000}]](https://tex.z-dn.net/?f=%5B%20%7B%5Crm%7B%5C%24%2010%2C000%20%2C%20%5C%24%2015%2C000%7D%5D)
, where b = $15000 and a = $10000.
The interval of the provided bidding is [$10,000,$12,000]. The probability is calculated as,
![\begin{array}{c}\\P\left( {X{\rm{ < 12,000}}} \right){\rm{ = }}1 - P\left( {X > 12000} \right)\\\\ = 1 - \int\limits_{12000}^{15000} {\frac{1}{{15000 - 10000}}} dx\\\\ = 1 - \int\limits_{12000}^{15000} {\frac{1}{{5000}}} dx\\\\ = 1 - \frac{1}{{5000}}\left[ x \right]_{12000}^{15000}\\\end{array}](https://tex.z-dn.net/?f=%5Cbegin%7Barray%7D%7Bc%7D%5C%5CP%5Cleft%28%20%7BX%7B%5Crm%7B%20%3C%2012%2C000%7D%7D%7D%20%5Cright%29%7B%5Crm%7B%20%3D%20%7D%7D1%20-%20P%5Cleft%28%20%7BX%20%3E%2012000%7D%20%5Cright%29%5C%5C%5C%5C%20%3D%201%20-%20%5Cint%5Climits_%7B12000%7D%5E%7B15000%7D%20%7B%5Cfrac%7B1%7D%7B%7B15000%20-%2010000%7D%7D%7D%20dx%5C%5C%5C%5C%20%3D%201%20-%20%5Cint%5Climits_%7B12000%7D%5E%7B15000%7D%20%7B%5Cfrac%7B1%7D%7B%7B5000%7D%7D%7D%20dx%5C%5C%5C%5C%20%3D%201%20-%20%5Cfrac%7B1%7D%7B%7B5000%7D%7D%5Cleft%5B%20x%20%5Cright%5D_%7B12000%7D%5E%7B15000%7D%5C%5C%5Cend%7Barray%7D)
![=1- \frac{[15000-12000]}{5000}\\\\=1-0.6\\\\=0.4](https://tex.z-dn.net/?f=%3D1-%20%5Cfrac%7B%5B15000-12000%5D%7D%7B5000%7D%5C%5C%5C%5C%3D1-0.6%5C%5C%5C%5C%3D0.4)
(b) The interval of the accepted bidding is [$10,000,$15,000], where b = $15,000 and a =$10,000. The interval of the given bidding is [$10,000,$14,000].
![\begin{array}{c}\\P\left( {X{\rm{ < 14,000}}} \right){\rm{ = }}1 - P\left( {X > 14000} \right)\\\\ = 1 - \int\limits_{14000}^{15000} {\frac{1}{{15000 - 10000}}} dx\\\\ = 1 - \int\limits_{14000}^{15000} {\frac{1}{{5000}}} dx\\\\ = 1 - \frac{1}{{5000}}\left[ x \right]_{14000}^{15000}\\\end{array} P(X14000)](https://tex.z-dn.net/?f=%5Cbegin%7Barray%7D%7Bc%7D%5C%5CP%5Cleft%28%20%7BX%7B%5Crm%7B%20%3C%2014%2C000%7D%7D%7D%20%5Cright%29%7B%5Crm%7B%20%3D%20%7D%7D1%20-%20P%5Cleft%28%20%7BX%20%3E%2014000%7D%20%5Cright%29%5C%5C%5C%5C%20%3D%201%20-%20%5Cint%5Climits_%7B14000%7D%5E%7B15000%7D%20%7B%5Cfrac%7B1%7D%7B%7B15000%20-%2010000%7D%7D%7D%20dx%5C%5C%5C%5C%20%3D%201%20-%20%5Cint%5Climits_%7B14000%7D%5E%7B15000%7D%20%7B%5Cfrac%7B1%7D%7B%7B5000%7D%7D%7D%20dx%5C%5C%5C%5C%20%3D%201%20-%20%5Cfrac%7B1%7D%7B%7B5000%7D%7D%5Cleft%5B%20x%20%5Cright%5D_%7B14000%7D%5E%7B15000%7D%5C%5C%5Cend%7Barray%7D%20P%28X%3C14%2C000%29%3D1-P%28X%3E14000%29)
![=1- \frac{[15000-14000]}{5000}\\\\=1-0.2\\\\=0.8](https://tex.z-dn.net/?f=%3D1-%20%5Cfrac%7B%5B15000-14000%5D%7D%7B5000%7D%5C%5C%5C%5C%3D1-0.2%5C%5C%5C%5C%3D0.8)
(c)
The amount that the customer bid to maximize the probability that the customer is getting the property is calculated as,
The interval of the accepted bidding is [$10,000,$15,000],
where b = $15,000 and a = $10,000. The interval of the given bidding is [$10,000,$15,000].
(d) The amount that the customer bid to maximize the probability that the customer is getting the property is $15,000, set by the seller. Another customer is willing to buy the property at $16,000.The bidding less than $16,000 getting considered as the minimum amount to get the property is $10,000.
The bidding amount less than $16,000 considered by the customers as the minimum amount to get the property is $10,000, and greater than $16,000 will depend on how useful the property is for the customer.
