Machine Part The lifetime of a machine part has a continuous...

Machine Part The lifetime of a machine part has a continuous distribution on the interval $(0,40)$ with probability density function $f,$ where $f(x)$ is proportional to $(10+x)^{-2}$ . Calculate the probability that the lifetime of the machine part is less than 6 Choose one of the following. Source: Society of Actuaries. $\begin{array}{lllllll}{\text { (a) } 0.04} & {\text { (b) } 0.15} & {\text { (c) } 0.47} & {\text { (d) } 0.53} & {\text { (e) } 0.94}\end{array}$

Machine Part The lifetime of a machine part has a continuous distribution on the interval $(0,40)$ with probability density function $f,$ where $f(x)$ is proportional to $(10+x)^{-2}$ . Calculate the probability that the lifetime of the machine part is less than 6 Choose one of the following. Source: Society of Actuaries.
$\begin{array}{lllllll}{\text { (a) } 0.04} & {\text { (b) } 0.15} & {\text { (c) } 0.47} & {\text { (d) } 0.53} & {\text { (e) } 0.94}\end{array}$

Key Concepts

Continuous Probability Distributions

This concept involves modeling a random variable whose values lie in a continuous range. In such distributions, probabilities are assigned to intervals rather than individual points, and they are completely described by a probability density function over the interval of interest.

Probability Density Function (PDF)

A PDF is a function that describes the relative likelihood of a continuous random variable taking on a particular value. It must be non-negative everywhere and integrate to 1 over its domain, ensuring that it properly represents a probability distribution.

Normalization of the PDF

When a PDF is given up to a constant of proportionality, the actual function must be normalized. This involves finding a constant such that the integral of the PDF over its entire domain equals 1. This step is crucial to establish that the function represents a valid probability distribution.

Cumulative Distribution Function (CDF) and Integration

The CDF represents the probability that the random variable takes a value less than or equal to a particular number. It is obtained by integrating the PDF from the lower bound of the distribution to the value of interest. This integration process is used to calculate probabilities for events defined over continuous intervals.

Transcript

00:01 All right, we're looking for the probability that the lifetime of machine is less than six.

00:05 We're given that we have an open interval of 0 to 40 for the lifetime of the machine, and that f of x is equal to quantity 10 plus x to the negative second.

00:23 Because it's said that it was proportional, that means we're going to have a constant, so we'll call it k, out in front.

00:29 That's the proportional part.

00:30 We're going to need to solve for k to figure out what that is.

00:34 And then we will need to figure out, again, 6 being the key number, for it being less than 6.

00:41 So to start with, we're going to integrate from 0 to 40 to find the overall the whole of the machine, which is going to then be 100 % or equal to 1.

00:52 So we're going to go from 0 to 40 of our function, and that needs to be equal to 1.

01:06 And again, we'll put our constant here outside for now.

01:10 So we're going to take the anti -derivative of our 10 plus x to the negative second.

01:16 So that'll be negative 10 plus x to the negative first.

01:24 And our constant k is still there.

01:29 And we then need to plug in 0 and 40.

01:35 And again, still going to be equal to 1 in the end.

01:39 So we can solve for k.

01:39 This is going to give us a negative 1 over 50 minus a negative 1 over 10, and that still gets multiplied by k, all still equal to 1.

02:03 Rearranging this because we have two negatives, we'll end up with 1 tenth minus 150th with k equals to 1.

02:21 So by getting a common denominator here, we would end up with 5 over 50 minus 1 over 50.

02:31 So that's going to be 4 over 50 k equals 1...

Please give Ace some feedback