Question

The Department of Transportation (DOT) monitors sealed bids for new road construction. For new access roads in a certain state, let low bid (thousands of dollars) and let estimate of fair cost of building the road (thousands of dollars). The joint probability density of X and Y is f(x,y)= e^-y/10/10y 0< y

Answer 1

Complete Question

The complete question is shown on the first and second uploaded image

Answer:

The marginal density function of Y is
`f(y)= \frac{e^{(-y)/(10) }}{10}`

The distribution is exponential distribution

and the expected value is
`E[Y] = 10`

Explanation:

From the question we are told that the function is

`f(x,y) = \frac{e^{-(y)/(10) }}{10y} \ \ \ 0< y <x<2y`

Now the marginal density function of Y i.e f(y) is mathematically evaluated by obtaining the probability density function of y as follows

`= \int\limits^(2y)_(y) { \frac{e^{(-y)/(10) }}{10y} } \, dx`

`= \frac{e^{(-y)/(10) }}{10y} * (2y - y )`

`= \frac{e^{(-y)/(10) }}{10}`

The distribution of the function above is exponential distribution with a rate parameter equals to

`\lambda = (1)/(10)`

The mean DOT estimate E{Y} is mathematically evaluated as

`E[Y] = (1)/(\lambda)`

substituting value

`E[Y] = (1)/((1)/(10) )`

`E[Y] =10`