Question

A student uses pens whose lifetime is an exponential random variable with mean 1 week. Use the central limit theorem to determine the minimum number of pens he should buy at the beginning of a 15-week semester, so that with probability .99 he does not run out of pens during the semester.

Answer 1

Answer:

Student needs pens= n = 27.04

Rounding off with upper floor function ⇒ n =28

Rounding off with lower floor function ⇒ n =27

Step-by-step explanation:

Given that lifetime of each pen is a exponential random variable with mean 1 week.

Let $S_{n}$ be total sum of lifetime of n pens.

So mean of $S_{n}$ = μ = n.1

Standard deviation of $S_{n}$ =$\sigma=\sqrt{n}$

Probability that he doesnot run out of pens= 0.99

Considering Sn be sum of n lifetimes, using central limit theorem

$\frac{S_{n}-n}{\sqrt{n}}\approx N(0,1)\\\\P(S_{n}>15)=[P(\frac{S_{n}-n}{\sqrt{n}})>\frac{15-n}{\sqrt{n}}]\\ 1-\phi(\frac{15-n}{\sqrt{n}})=\phi(-(\frac{15-n}{\sqrt{n}}))=0.99$

From table of standard normal distribution

$\frac{15-n}{\sqrt{n}}=-2.3263\\15-n=-2.3263\sqrt{n}\\n-2.3263-15\sqrt{n}$

Solving the quadratic Equation in variable x we get

n=27.04