Suppose the time interval between two consecutive defective light bulbs from a production line has a uniform distribution over a

Question

OverLord2011

6 days ago

8

Suppose the time interval between two consecutive defective light bulbs from a production line has a uniform distribution over a

n interval from 0 to 90 minutes. a) What is the mean of the time interval? (2 points) b) What is the probability that the time interval between two consecutive defective light bulbs will be between 10 and 35 minutes? (6 points) c) What is the standard deviation of the time interval? (2 points) d) W hat is the probability that the time interval between two consecutive defective light bulbs will be less than 10 minutes? (4 points)

Mathematics

1 answer:

lawyer [4K] · Answer 1 · 2026-02-27T21:47:45+03:00

Answer:

a) The average of the time interval is 45 minutes.

b) There is a 27.78% chance that the time gap between two successive defective light bulbs will lie between 10 and 35 minutes.

c) The standard deviation for the time duration is 25.98 minutes.

d) There is an 11.11% likelihood that the time interval between two successive defective light bulbs will be under 10 minutes.

Step-by-step explanation:

A uniform probability situation occurs when all outcomes have an equal chance of happening.

In this context, we identify a lower and upper limit for the distribution known as 'a' and 'b', respectively.

The likelihood of finding a value X that is less than x is determined by this formula.

$P(X < x) = \frac{x - a}{b-a}$

The likelihood of obtaining a value X that lies between c and d is expressed as:

$P(c \leq X \leq d) = \frac{d-c}{b-a}$

The average of the uniform distribution is:

$M = \frac{a+b}{2}$

The standard deviation of the uniform distribution can be calculated as:

$S = \sqrt{\frac{(b-a)^{2}}{12}}$

The uniform distribution covers an interval from 0 to 90 minutes.

This indicates that $a = 0, b = 90$

a) What is the mean of the time interval?

$M = \frac{a+b}{2} = \frac{0 + 90}{2} = 45$

The mean of the time interval is 45 minutes.

b) What is the probability that the time gap between two consecutive defective light bulbs will be between 10 and 35 minutes?

$P(10 \leq X \leq 35) = \frac{35-10}{90-0} = 0.2778$

There is a 27.78% chance that the time interval between two successive defective light bulbs falls between 10 and 35 minutes.

c) What is the standard deviation of the time interval?

$S = \sqrt{\frac{(90-0)^{2}}{12}} = 25.98$

The standard deviation of the time interval is 25.98 minutes.

d) What is the probability that the time interval between two consecutive defective light bulbs will be less than 10 minutes?

$P(X < 10) = \frac{10 - 0}{90 - 0} = 0.1111$

There is an 11.11% likelihood that the time duration between two consecutive defective light bulbs will be shorter than 10 minutes.