The lifetime, in years, of a certain type of pump is a...

The lifetime, in years, of a certain type of pump is a random variable with probability density function egin{equation*} f(x) = egin{cases} frac{64}{(x+2)^3} & x > 0\0 & x leq 0 end{cases} end{equation*} a. What is the probability that a pump lasts more than two years? b. What is the probability that a pump lasts between two and four years? c. Find the mean lifetime. d. Find the variance of the lifetimes. e. Find the cumulative distribution function of the lifetime. f. Find the median lifetime. g. Find the 60th percentile of the lifetimes.

Transcript

00:01 In this video, we are going to focus on solving many different parts of this very question based on the given probability density function.

00:07 So the probability density function is f of x is equal to 64 divided by x plus 2 raised to the power.

00:15 It will be here 5.

00:17 And here the condition on x is between infinity to 0.

00:22 Now, if we proceed next, it can be said that x denotes lifetime of the power.

00:30 So it will be here lifetime of the pump.

00:36 Here we have to understand that this x is the random variable.

00:40 Now here we can say, so before moving next, we can here write that this very f of x value is equal to 0 for the case when x value is basically less than 0.

00:53 So when it is less than equal to 0, it is 0 and when the value of x is greater than 0 or we can say between 0 and infinity, it will be.

01:00 This very expression.

01:01 So in the first part of this very problem, we have to find the probability that pump lasts more than two years.

01:07 So probability of x greater than two will be here equals to integration from 2 to infinity and then here it is f of x times d x.

01:16 So this will be here equals to 64 and then here we can write that it will be infinity here it is 2 and then it is 1 divided by x plus 2 raise to the power 5.

01:27 So it will be here d x.

01:30 Now, if we solve this very problem again, or we can say if we simplify it, we are going to get the value as minus 16 divided by.

01:38 It will be here, x plus, and then here we have to raise to the power 4, and then it will be here from 2 to infinity.

01:48 So here it can be written that this very value is going to come around 1 divided by 16, which is approximately equals to, we can write its value which is going to be 0 .0 .0.

02:00 625.

02:01 So we have got the value of probability and now here we can say that let us solve the second part of this very question because first part answer that is the probability that a pump lasts more than two years is given by 0 .0625.

02:19 So let's solve the third part.

02:22 So in the sorry, let's solve the second part.

02:25 So in the second part we have to find the probability that a pump lasts between 2 .2 .5.

02:30 To four years.

02:30 So basically we have to find out the value of this very expression and hence let us see how we can do so.

02:37 It will be here four and this is here equals to integration from 2 to 4 and then it will be here f of x times dx and then we are going to get the value of this very expression as integration from 2 to 4 and then it will be here 64 1 divided by x plus 2 raise to the power 5 and then it is here dx.

02:58 So if it is simplified, it will be minus 16 divided by x plus 2 and then it is here raised to the power 4 and then it will be between 2 to 4.

03:11 So from this very information we can say it is going to come around 0 .0504...

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