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:

[tex](a)\ c = \frac{1}{15}[/tex]

[tex](b)\ 40\%[/tex]

[tex](c)\ 60\%[/tex]

Step-by-step explanation:

The given parameters can be represented as:

[tex]P_Y(y) \ge 0, y =1,2,3,4,5[/tex]

[tex]P_y(y) = cy, y=1,2,3,4,5[/tex]

Solving (a): The value of c

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

[tex]\sum\limit^5_{y=1}P_Y(y_i) = 1[/tex]

Given that:

[tex]P_y(y) = cy, y=1,2,3,4,5[/tex]

This is translated to:

[tex]c*1 + c * 2 + c * 3 + c * 4 + c * 5 = 1[/tex]

[tex]c + 2c + 3c + 4c + 5c = 1[/tex]

[tex]15c = 1[/tex]

Solve for c

[tex]c = \frac{1}{15}[/tex]

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

This implies that: y = 1,2,3

So, we make use of:

[tex]P(Y \le 3) = P(Y=1) + P(y=2) + P(Y=3)[/tex]

Recall that:

[tex]P_y(y) = cy, y=1,2,3,4,5[/tex]

Substitute [tex]c = \frac{1}{15}[/tex]

[tex]P_y(y) =\frac{1}{15}y[/tex]

So:

[tex]P(Y \le 3) = P(Y=1) + P(y=2) + P(Y=3)[/tex]

[tex]P(Y\le 3) = \frac{1}{15} * 1 +\frac{1}{15} * 2 +\frac{1}{15} * 3[/tex]

[tex]P(Y\le 3) = \frac{1}{15} +\frac{2}{15} +\frac{3}{15}[/tex]

Take LCM

[tex]P(Y\le 3) = \frac{1+2+3}{15}[/tex]

[tex]P(Y\le 3) = \frac{6}{15}[/tex]

[tex]P(Y\le 3) = 0.4[/tex]

Express as percentage

[tex]P(Y\le 3) = 0.4*100\%[/tex]

[tex]P(Y\le 3) = 40\%[/tex]

(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:

[tex]P(2 \le Y \le 4) = P(Y=2) + P(Y=3) + P(Y=4)[/tex]

[tex]P(2 \le Y \le 4) = 2 * \frac{1}{15} + 3 * \frac{1}{15} + 4 * \frac{1}{15}[/tex]

[tex]P(2 \le Y \le 4) = \frac{2}{15} + \frac{3}{15} + \frac{4}{15}[/tex]

Take LCM

[tex]P(2 \le Y \le 4) = \frac{2+3+4}{15}[/tex]

[tex]P(2 \le Y \le 4) = \frac{9}{15}[/tex]

[tex]P(2 \le Y \le 4) = 0.6[/tex]

Express as percentage

[tex]P(2 \le Y \le 4) = 0.6 * 100\%[/tex]

[tex]P(2 \le Y \le 4) = 60\%[/tex]

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