Question

A contractor is required by a county planning department to submit 1, 2, 3, 4, or 5 forms (depending on the nature of the project) when applying for a building permit. Let y denote the number of forms required for an application, and suppose the mass function is given by p(y) 5 cy for y 5 1, 2, 3, 4, or 5. Determine the value of c, as well as the long-run proportion of applications that require at most three forms and the long-run proportion that require between two and four forms, inclusive.

Answer 1

`(a)\ c = (1)/(15)`

`(b)\ 40\%`

`(c)\ 60\%`

Explanation:

The given parameters can be represented as:

`P_Y(y) \ge 0, y =1,2,3,4,5`

`P_y(y) = cy, y=1,2,3,4,5`

Solving (a): The value of c

To do this, we make use of the following rule;

`\sum\limit^5_(y=1)P_Y(y_i) = 1`

Given that:

`P_y(y) = cy, y=1,2,3,4,5`

This is translated to:

`c*1 + c * 2 + c * 3 + c * 4 + c * 5 = 1`

`c + 2c + 3c + 4c + 5c = 1`

`15c = 1`

Solve for c

`c = (1)/(15)`

(b) The proportions of applications that requires at most 3 forms

This implies that: y = 1,2,3

So, we make use of:

`P(Y \le 3) = P(Y=1) + P(y=2) + P(Y=3)`

Recall that:

`P_y(y) = cy, y=1,2,3,4,5`

Substitute
`c = (1)/(15)`

`P_y(y) =(1)/(15)y`

So:

`P(Y \le 3) = P(Y=1) + P(y=2) + P(Y=3)`

`P(Y\le 3) = (1)/(15) * 1 +(1)/(15) * 2 +(1)/(15) * 3`

`P(Y\le 3) = (1)/(15) +(2)/(15) +(3)/(15)`

Take LCM

`P(Y\le 3) = (1+2+3)/(15)`

`P(Y\le 3) = (6)/(15)`

`P(Y\le 3) = 0.4`

Express as percentage

`P(Y\le 3) = 0.4*100\%`

`P(Y\le 3) = 40\%`

(c) The proportions of applications that requires between 2 and 4 forms (inclusive)

This implies that: y = 2,3,4

So, we make use of:

`P(2 \le Y \le 4) = P(Y=2) + P(Y=3) + P(Y=4)`

`P(2 \le Y \le 4) = 2 * (1)/(15) + 3 * (1)/(15) + 4 * (1)/(15)`

`P(2 \le Y \le 4) = (2)/(15) + (3)/(15) + (4)/(15)`

Take LCM

`P(2 \le Y \le 4) = (2+3+4)/(15)`

`P(2 \le Y \le 4) = (9)/(15)`

`P(2 \le Y \le 4) = 0.6`

Express as percentage

`P(2 \le Y \le 4) = 0.6 * 100\%`

`P(2 \le Y \le 4) = 60\%`