A municipal bus company has started operations in a new subdivision. Records were kept on the nu

step 1 Welcome to Numerid. In the current problem, we are given this values of X, which are varying fro

A municipal bus company has started operations in a new...

Key Concepts

Variance and Standard Deviation of a Discrete Random Variable

The variance measures the spread of the distribution by computing the average squared deviation from the mean, and the standard deviation is the square root of the variance, providing a measure of dispersion in the same units as the original variable. These statistics assess the reliability and variability of the expected outcome.

Mean (Expected Value) of a Discrete Random Variable

The mean or expected value of a discrete random variable is a measure of the central tendency, calculated by summing the products of each outcome and its probability. It provides a long-run average outcome if the experiment were repeated many times.

Transformation of Random Variables

When a random variable is transformed by applying a linear function, such as multiplying the number of riders by a fixed cost per ride, the mean and variance of the new variable change in predictable ways. Specifically, the mean is scaled by the constant and the variance is scaled by the square of the constant, which helps in analyzing outcomes related to cost, revenue, or other derived quantities.

Probability Calculation over a Range of Outcomes

Calculating the probability of events that are defined by a range of outcomes involves summing the probabilities for all individual outcomes within that range. For example, finding the probability of having at least a certain number of riders requires adding the probabilities for all outcomes that meet or exceed that number.

Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) of a discrete random variable gives the probability that the variable takes a value less than or equal to a specific value. It is constructed by summing the probabilities of all outcomes up to and including the target outcome, and it is useful for answering probability questions about ranges or intervals.

Independence and Joint Probability

When dealing with independent events, such as separate days of bus service, the joint probability of two independent events occurring simultaneously is the product of their individual probabilities. This principle is essential when calculating the likelihood of events occurring on multiple independent trials.

Graphical Representation of Distributions

Graphing a discrete probability distribution usually involves creating a bar graph where the horizontal axis represents the discrete outcomes and the vertical axis represents the corresponding probabilities. This visual representation helps in quickly understanding the distribution and identifying the most and least likely outcomes.

Discrete Probability Distribution

A discrete probability distribution is a function that provides the probabilities of occurrence of different possible outcomes in a discrete sample space where the total probability sums to one. It is typically represented by a list or table mapping each outcome to its probability, and it is used to model situations where the outcomes can be counted, such as the number of bus riders in this scenario.

Transcript

00:01 Welcome to numerid.

00:02 In the current problem, we are given this values of x, which are varying from 22 and 27, and then the corresponding probabilities which adds up to 1.

00:15 Okay.

00:16 And then we are going to find the capital px, that is the probability of x less than equals to small x, the cumulative probability.

00:27 So what will it be? x less than equals to 20 is same.

00:31 Value 0 .02.

00:34 This x less than equals to 21 is x equals to 20 x equals to 21.

00:40 So we add them and get a 0 .14.

00:44 And then next we get 0 .37 and then we get 0 .68.

00:53 That is what is the probability of x less than equals to 23? it is x equals to 20 plus x equals to 21 plus x equals to 2.

01:01 22 plus x equals to 23 so going by that we get 0 .87 0 .95 0 .5 sorry 9 898 and then 1 and this two numbers should always match okay because this is the total of all the values now we have to draw the diagrams so for drawing the diagrams let us start by drawing the pair of axis so we start by 20 21 now if you see the highest value over here is 0 .31 correct so i can take 0 .3 0 .3 0 .2 0 .1 okay so how do i have first is 0 .2 which is really nearby so it this is 0 .02 then 0 .14 which is little more than this correct sorry 0 .12 this one so like this next is 0 .23 so it is little higher than this so next is 0 .31 which is this highest end next 1 -9 so it is little less than 0 .2 right so it is little less than then 0 .08 that is little less than 0 .3 correct and the last one is 0 .2 okay so this is the probability mass function now we will have the cumulative mass function or cumulative probability distribution so oops again 20 21 22 23 24 25 25 26 27 so this is where it will touch 1 right and it will continue oops that's a not diagram it will continue to stay at 1 so this is the level of 1 so clearly this should be half way 0 .5 this should be 0 .75 this should be 0 .25 so let's see who lies where first thing is 0 .02 which is like this low the moment it touches 21 the probability ends up at 0 .14 so this is where in the half way you can think of that so i'm giving you a of this okay and then again it continues to go there okay at that level then 0 .37 where is 0 .37 midway of this so it goes till here it is 0 .37 now now 0 .68 again till here after that 0 .87 oh that's quite a high after that 0 .95 now, 0 .95 is almost like a, you know, i'm reaching there and then 0 .98, almost one.

05:14 Oops, look at least here.

05:17 This is where the level of 1.

05:19 So, it's a free hand drawing.

05:20 I hope you understand.

05:21 You are not considering the first line that i had drawn.

05:24 This is where the line goes, right? so, this is the cumulative probability distribution.

05:37 This is also known as step diagram.

05:42 Now, so these are part a and part b.

05:48 Now for part c, we have that what is the probability that in a randomly chosen weekday, there will be at least 24 riders from the subdivision of the service.

05:58 That means at least probability at least 24.

06:08 That means x should be at least 24.

06:10 So, x greater than equals to 24.

06:13 That means probability of 24 25 26 27 so if you add these okay so where is it if you add this this this and this you get 0 .32 okay so this is the answer for the third sub part now next is it is it is asked 2 week days are chosen at random what is the probability that there will be fewer than 23 riders correct so if i think of 23 riders how less than fewer so probability oops that two riders okay each less than 23 correct so less than 23 means this part correct now if you see these are independent events so choosing one wouldn't have impact on another one so what will happen x less than 23 two times why because two riders so means by this what i mean if i have one day that has 23 riders or 25 riders for example that wouldn't suffice or imply you that you know the other day can or cannot have 25 riders.

08:13 So, what do i mean by independence? that one event, the occurrence of one event is not influencing the other event.

08:22 Now what is dependent event in that case? for example in this one red, one black, one white, one green colored.

08:31 Things were there now if you take out one black item then what will happen again occurrence of another black item will be defined what to say dependent on this event but if you think of this you if you take one day out of the entire week and if that has 24 riders doesn't mean that it will impact other day from happening 24 riders, correct? the way things are getting changed here.

09:06 So, in this case, now what is the probability of x less than 23? it is probability x less than equals to 20, correct? so, what is probability x less than equals to 22? it is sum of these values, correct? so, we have 0 .37 pole square, which is equal to 0 .1369.

09:39 Now, the next part that asks is that find them, like, we have to find the mean and sd, okay, from this distribution.

09:52 So, for a mean, the formula is x into px...

Please give Ace some feedback