Question

In October , Apple introduced a much smaller variant of the Apple iPad, known as the iPad Mini. Weighing less than ounces, it was about lighter than the standard iPad. Battery tests for the iPad Mini showed a mean life of hours (The Wall Street Journal, October , ). Assume that battery life of the iPad Mini is uniformly distributed between and hours.a. Give a mathematical expression for the probability density function of battery life.A.B.C.The correct answer is:- Select your answer -b. What is the probability that the battery life for an iPad Mini will be hours or less (to 4 decimals)

Answer 1

a. Probability density function for the battery life

`f(x)={\begin{cases}{\frac {1}{12-8.5}}=(1)/(3.5)&\mathrm {for} \ 8.5\leq x\leq 12,\\[8pt]0&\mathrm {for} \ x<8.5\ \mathrm {or} \ x>12\end{cases}}`

b. P(x<11) = 0.7143

Explanation:

The question is incomplete

In October, Apple introduced a much smaller variant of the Apple iPad, known as the iPad Mini. Weighing less than 11 ounces, it was about 50% lighter than the standard iPad. Battery tests for the iPad Mini showed a mean life of hours (The Wall Street Journal, October 31, 2012). Assume that battery life of the iPad Mini is uniformly distributed between 8.5 and 12 hours.

a. Give a mathematical expression for the probability density function of battery life.

b. What is the probability that the battery life for an iPad Mini will be 11 hours or less (to 4 decimals).

a. We model the battery life as random variable with an uniform distribution with parameters a (min. value)=8.5 hours and b (max. value)=12 hours.

The probability that the battery life takes any value between 8.5 and 12 is constant. Also, the probability that takes any value outside the interval (8.5, 12) is 0.

We can express that as:

`f(x)={\begin{cases}{\frac {1}{12-8.5}}=(1)/(3.5)&\mathrm {for} \ 8.5\leq x\leq 12,\\[8pt]0&\mathrm {for} \ x<8.5\ \mathrm {or} \ x>12\end{cases}}`

The last is the probability density function for the battery life.

b. We can calculate P(x<11) using the cumulative density function or integrating the density function between x=0 (or x=a=8.5) and x=11.

`P(x<11)=\int\limits^(11)_(8.5) (1)/(3.5) \, dx=(1)/(3.5)(x_2-x_1)=(1)/(3.5)(11-8.5)=(2.5)/(3.5)= 0.7143`