An automobile dealer calculates the proportion of new cars...

An automobile dealer calculates the proportion of new cars sold that have been returned a various numbers of times for the correction of defects during the warranty period. The results are shown in the following table. $$ \begin{array}{lccccc} \hline \text { Number of returns } & 0 & 1 & 2 & 3 & 4 \\ \hline \text { Proportion } & 0.28 & 0.36 & 0.23 & 0.09 & 0.04 \\ \hline \end{array} $$ a. Graph the probability distribution function. b. Calculate and graph the cumulative probability distribution. c. Find the mean of the number of returns of an automobile for corrections for defects during the warranty period. d. Find the variance of the number of returns of an automobile for corrections for defects during the warranty period.

An automobile dealer calculates the proportion of new cars sold that have been returned a various numbers of times for the correction of defects during the warranty period. The results are shown in the following table.
$$
\begin{array}{lccccc}
\hline \text { Number of returns } & 0 & 1 & 2 & 3 & 4 \\
\hline \text { Proportion } & 0.28 & 0.36 & 0.23 & 0.09 & 0.04 \\
\hline
\end{array}
$$
a. Graph the probability distribution function.
b. Calculate and graph the cumulative probability distribution.
c. Find the mean of the number of returns of an automobile for corrections for defects during the warranty period.
d. Find the variance of the number of returns of an automobile for corrections for defects during the warranty period.

Key Concepts

Expected Value (Mean)

The mean or expected value of a discrete random variable represents the long-run average outcome when the experiment is repeated many times. It is calculated as the sum of the products of each outcome and its associated probability, serving as a measure of central tendency.

Variance

Variance quantifies the spread or dispersion of the values of a discrete random variable around the mean by calculating the average of the squared differences between each outcome and the mean. It provides insight into the variability and consistency of the outcomes.

Cumulative Distribution Function (CDF)

The cumulative distribution function provides the probability that a discrete random variable takes on a value less than or equal to a given number. It is formed by the cumulative sum of the probabilities from the probability distribution function, offering a way to assess the distribution of probabilities up to any specified point.

Probability Distribution Function (PDF)

A probability distribution function for a discrete random variable specifies the probabilities associated with each possible outcome, ensuring that the sum of these probabilities equals 1. It provides the fundamental description of how probability is distributed among the various discrete outcomes.

Transcript

00:01 Welcome to numerate in the current problem.

00:05 We are given the automobile, the number of automobiles that are being returned.

00:12 So, we can write over here that the number of automobiles that are being returned is denoted by x, okay? and we are given x can be 0, 1, 2, 3 and 4.

00:29 And the probabilities of x assuming those values are given to be 0 .28 0 .36 0 .23, 0 .0 .0 .0 .0 .0 .0 .0 .0.

00:45 0 .0 .0.

00:48 Now, the first thing that we check, this adds up to 1.

00:51 That means we can accept it to be a probability distribution.

00:59 Is to draw a probability distribution for this.

01:02 So, if i plot here, the values 0 ,1, 2, 3, 4, and if we choose the scales correctly, okay? so, if you see the highest value is 0 .36.

01:24 So, somewhere here, if we take 0 .4, so 0 .28 will be somewhere over here.

01:34 And then if we see 0 .36, that will go around here.

01:43 So, correct.

01:46 The drawings are not accurate enough because, you know, i'm drawing freehand.

01:49 I hope you a little understand that there.

01:53 Then 0 .23 that will go further down.

01:56 So, it is here in 0 .9, which will really be low over here and 0 .04 will be again very low.

02:04 So, this is how px the probability mass function would look like, right? it's a discrete distribution, so it will be pmf.

02:16 Now, they're also asking you to find the cumulative distribution function, correct? so, we define that with capital p and we denote the notation by this.

02:27 So, we have this probability same here.

02:30 What is the probability that x is less than equal to 1? x is less than equals to 1 means it can be 0 or 1 so we add these two probabilities so if we go ahead then we have 0 .64 now probability x less than equals to now 2 correct so x when will oops so when will x be less than equal to 2 when it is 2 1 or 0 so we have to add these 3 or simply the add the sum of these 2 is written here so we can just add these two numbers okay that's a faster calculation when you don't have a calculator nearby so it will be 0 .87 okay and then zero see going forward with same logic we get 0 .96 and 1 okay and see always this would add up to 1 now they're asking so this was part a okay this is for part b also the diagram so how do we get the diagram before we go ahead and do that i will rather have the diagram over here.

03:39 Okay.

03:40 So, here the highest value will be one, correct? the highest value.

03:44 And if you see all these values keeps on increasing, correct? so, so, oops, we will see an upward trend over here.

04:02 So, 1, 2, 3, 4.

04:06 So the first one will start at 0 .28, okay? and then it will continue going like this and at this, this will go up at 0 .64.

04:22 So quite a good up.

04:27 And then 0 .64, then 0 .87.

04:32 So it's near about 9.

04:33 Correct.

04:34 So it will be very close proximity to 9.

04:37 And then it is really touching 0 .96.

04:41 So it is going like this...