SaiBook.us Statistics

Lesson Example Discussion Quiz: Class Homework

Example

Title: Continuous random variable

Grade: 9-a Lesson: S4-L7

Explanation: The best way to understand statistics is by looking at some examples. Take turns and read each example for easy understanding.

Examples:

The amount of bread (in hundreds of pounds) x that a certain bakery is able to sell in a day is found to be a numerical valued random phenomenon, with a probability function specified by the probability density function f (x) is given by

$f(x) = \begin{cases} Ax,for & 0≤x≤10 \\ A(20-x),for & 10≤x≤20 \\ 0,& otherwise \\ \end{cases}$

(a) Find the value of A.
(b) What is the probability that the number of pounds of bread that will be sold tomorrow is
(i) More than 10 pounds
(ii) Less than 10 pounds
(iii)between 5 and 15 pounds?

Step 1a

Finding the value of 'A'

Explanation: \$\int_-\infty^\infty\$f(x)dx = 1,

x is a random variable.

\$\int_0^10\$Axdx + \$\int_10^20\$A(20-x)dx = 1

\$A(x^2/2)_0^10 +(20x - x^2/2)_10^20\$ = 1

A[(50-0) + (400-200) - (200-50)] = 1

\$A = 1/100\$

Step 1b

Determine Probability that the number of pounds of bread that will be sold tomorrow is more than 10 pounds

Explanation:

Probability that the number of pounds of bread that will be sold tomorrow is more than 10 pounds is denoted by P(10≤x≤20)

P(10≤x≤20) = \$\int_10^20 1/100(20-x)dx\$

P(10≤x≤20) = \$1/100 (20x -x^2/2)_10^20\$

P(10≤x≤20) = \$1/100 (20(20-10) - (20-10)^2/2)\$

P(10≤x≤20) = \$ 1/100 (20(10) - 10^2/2)\$

P(10≤x≤20) = \$ 1/100 (200 - 100/2)\$

P(10≤x≤20) = 0.5

Step 1c

Determine Probability that the number of pounds of bread that will be sold tomorrow is less than 10 pounds

Explanation:

Probability that the number of pounds of bread that will be sold tomorrow is less than 10 pounds is denoted by P(0≤x≤10)

P(0≤x≤10) = \$\int_0^10 1/100 xdx\$

P(0≤x≤10) = \$1/100 (x^2/2)_0^10\$

P(0≤x≤10) = \$1/100 ((10-0)^2/2)\$

P(0≤x≤10) = \$1/100 (10^2/2)\$

P(0≤x≤10) = \$1/100 (100/2) \$

P(0≤x≤10) = 0.5

Step 1d

Determine Probability that the number of pounds of bread that will be sold tomorrow is between 5 and 15 pounds

Explanation:

Probability that the number of pounds of bread that will be sold tomorrow is between 5 and 15 pounds is denoted by P(5≤x≤15)

P(5≤x≤15) = \$\int_5^10 1/100 xdx\$ + \$\int_10^15 1/100(20-x)dx\$

P(5≤x≤15) = \$1/100 (x^2/2)_5^10 + 1/100 (20x -x^2/2)_10^15\$

P(5≤x≤15) = \$1/100 ((10-5)^2/2) + 1/100 (20(15-10) - (15-10)^2/2)\$

P(5≤x≤15) = \$1/100 (5^2/2) + 1/100 (20(5) - 5^2/2)\$

P(5≤x≤15) = \$1/100 (25/2) + 1/100 (100 - 25/2)\$

P(5≤x≤15) = 0.75