Question

At an ocean-side nuclear power plant, seawater is used as part of the cooling system. This raises the temperature of the water that is discharged back into the ocean. The amount that the water temperature is raised has a uniform distribution over the interval from 10 to 25�C. (a) What is the probability that the temperature increase will be (1) less than 20 degrees C? (2) between 20 and 22C?(b) Suppose that a temperature increase of more than 18 degrees C is considered to be potentially harmful to the environment. What is the probability, at any point of time, that the temperature increase is potentially dangerous? (c) what is the expected value of the temperature increase?

Answer 1

Concepts and reason

Uniform distribution: Uniform distribution is a continuous probability distribution. It is defined between two parameters A and B. Here, A is called minimum value and B is called maximum value. Let X be continuous random variable with uniform distribution $U\left( {A,B} \right)$ . That is, $X \sim U\left( {A,B} \right)$ . Moreover, the probability density function for X is,

${f_X}\left( x \right) = \frac{1}{{B - A}}{\rm{ }}A < x < B$ .

Fundamentals

• Formula for finding the value of $P\left( {X > x} \right)$ is $P\left( {X > x} \right) = \int\limits_x^B {f\left( x \right)dx}$ .

• Formula for finding the value of $P\left( {X < x} \right)$ is $P\left( {X < x} \right) = \int\limits_A^x {f\left( x \right)dx}$ .

• Formula for finding the value of $P\left( {X \ge x} \right)$ is $P\left( {X \ge x} \right) = \int\limits_x^B {f\left( x \right)dx}$ or $P\left( {X \ge x} \right) = 1 - P\left( {X < x} \right)$ .

• Formula for finding the value of $P\left( {a \le X \le b} \right)$ is $P\left( {a \le X \le b} \right) = \int\limits_a^b {f\left( x \right)\,\,\,dx}$

• Formula for mean of X is $E\left( X \right) = \frac{{B + A}}{2}$ .

(a.1)

The probability that the temperature increase will be less than 20 degrees C is obtained as shown below:

From information given, the amount that the water temperature is raised has a uniform distribution over the interval from 10 C to 25 C. The probability density function is,

$\begin{array}{c}\\{f_X}\left( x \right) = \frac{1}{{B - A}}{\rm{ ; }}A < x < B\\\\ = \frac{1}{{25 - 10}}\\\\ = \frac{1}{{15}}\\\\{f_X}\left( x \right) = 0.0667\,\,\,\,\,\,\,\,\,\,\,\,\,\,;\,\,\,\,10 < x < 25\\\end{array}$

The required probability is,

$\begin{array}{c}\\P\left( {X < 20} \right) = \int\limits_{10}^{20} {f\left( x \right)dx} \\\\ = \int\limits_{10}^{20} {0.0667\,dx} \\\\ = 0.0667\int\limits_{10}^{20} {1\,dx} \\\\ = 0.0667\left[ x \right]_{10}^{20}\\\end{array}$

$\begin{array}{c}\\ = 0.0667\left[ {20 - 10} \right]\\\\ = 0.0667 \times 10\\\\ = 0.667\\\end{array}$

(a.2)

The probability that the temperature increase will be between 20 C and 22 C is obtained as shown below:

The required probability is,

$\begin{array}{c}\\P\left( {20 \le X \le 22} \right) = \int\limits_{20}^{22} {f\left( x \right)\,\,\,dx} \\\\ = \int\limits_{20}^{22} {0.0667\,\,\,dx} \\\\ = 0.0667\int\limits_{20}^{22} {1\,\,\,dx} \\\\ = 0.0667\left[ x \right]_{20}^{22}\\\end{array}$

$\begin{array}{c}\\ = 0.0667\left[ {22 - 20} \right]\\\\ = 0.0667 \times 2\\\\ = 0.1334\\\end{array}$

(b)

The probability that temperature increase of more than 18 degrees C is considered to be potentially harmful to the environment is obtained as shown below:

The required probability is,

$\begin{array}{c}\\P\left( {X > 18} \right) = \int\limits_{18}^{25} {f\left( x \right)\,\,\,dx} \\\\ = \int\limits_{18}^{25} {0.0667\,\,\,dx} \\\\ = 0.0667\int\limits_{18}^{25} {1\,\,\,dx} \\\\ = 0.0667\left[ x \right]_{18}^{25}\\\end{array}$

$\begin{array}{c}\\ = 0.0667\left[ {25 - 18} \right]\\\\ = 0.0667 \times 7\\\\ = 0.4669\\\end{array}$

The probability value is 0.4669 which is reasonably high; this implies that an increase of more than 18 degrees C in temperature is considered to be potentially harmful to the environment.

(c)

The expected value of the temperature increase is obtained as shown below:

$\begin{array}{c}\\E\left( X \right) = \frac{{B + A}}{2}\\\\ = \frac{{25 + 10}}{2}\\\\ = \frac{{35}}{2}\\\\ = 17.5\\\end{array}$

Ans: Part a.1

The probability that he temperature increase will be less than 20 degrees C is 0.667.

Part a.2

The probability that the temperature increase will be between 20 C and 22 C is 0.1334.

Part b

The probability that temperature increase of more than 18 degrees C is considered to be potentially harmful to the environment is 0.4669.

Part c

The expected value of the temperature increase is 17.5 degrees C.