Sure, let's break down the solution step-by-step.
### Part (a)
Question: Give a mathematical expression for the probability density function of battery life.
Solution:
Since the battery life of the iPad Mini is uniformly distributed between 8.5 and 12 hours, we use the uniform distribution formula for the probability density function (pdf), which is:
[tex]\[ f(x) = \frac{1}{b - a} \][/tex]
where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are the lower and upper bounds of the distribution, respectively. Here, [tex]\( a = 8.5 \)[/tex] and [tex]\( b = 12 \)[/tex].
[tex]\[ f(x) = \frac{1}{12 - 8.5} = \frac{1}{3.5} \][/tex]
Thus, the pdf is:
[tex]\[ f(x) = \begin{cases}
\frac{1}{3.5} & \text{for } 8.5 \leq x \leq 12 \\
0 & \text{elsewhere}
\end{cases} \][/tex]
The correct answer is:
[tex]\[ A. \, f(x) = \begin{cases}
\frac{1}{3.5} & \text{for } 8.5 \leq x \leq 12 \\
0 & \text{elsewhere}
\end{cases} \][/tex]
### Part (b)
Question: What is the probability that the battery life for an iPad Mini will be 10 hours or less?
Solution:
For a uniform distribution, the probability [tex]\( P(a \leq X \leq b) \)[/tex] can be found using the formula [tex]\( \frac{x - a}{b - a} \)[/tex]. Here, we want the probability that [tex]\( X \)[/tex] is 10 or less, with [tex]\( a = 8.5 \)[/tex] and [tex]\( b = 12 \)[/tex].
[tex]\[ P(X \leq 10) = \frac{10 - 8.5}{12 - 8.5} = \frac{1.5}{3.5} = 0.4286 \][/tex]
So, the probability is [tex]\( 0.4286 \)[/tex].
### Part (c)
Question: What is the probability that the battery life for an iPad Mini will be at least 11 hours?
Solution:
Again, for a uniform distribution, the probability [tex]\( P(X \geq x) \)[/tex] can be found using the formula [tex]\( \frac{b - x}{b - a} \)[/tex]. Here, we want the probability that [tex]\( X \)[/tex] is at least 11, with [tex]\( a = 8.5 \)[/tex] and [tex]\( b = 12 \)[/tex].
[tex]\[ P(X \geq 11) = \frac{12 - 11}{12 - 8.5} = \frac{1}{3.5} = 0.2857 \][/tex]
So, the probability is [tex]\( 0.2857 \)[/tex].
### Part (d)
Question: What is the probability that the battery life for an iPad Mini will be between 9.5 and 11.5 hours?
Solution:
For the range [tex]\( 9.5 \leq X \leq 11.5 \)[/tex], the probability can be found using:
[tex]\[ P(9.5 \leq X \leq 11.5) = \frac{11.5 - 9.5}{12 - 8.5} = \frac{2}{3.5} = 0.5714 \][/tex]
So, the probability is [tex]\( 0.5714 \)[/tex].
### Summary of Answers
a. [tex]\( f(x) = \begin{cases}
\frac{1}{3.5} & \text{for } 8.5 \leq x \leq 12 \\
0 & \text{elsewhere}
\end{cases} \)[/tex]
b. The probability that the battery life will be 10 hours or less is [tex]\( 0.4286 \)[/tex].
c. The probability that the battery life will be at least 11 hours is [tex]\( 0.2857 \)[/tex].
d. The probability that the battery life will be between 9.5 and 11.5 hours is [tex]\( 0.5714 \)[/tex].
