Given the probability distribution function: x 0 1 2 Probability 0.25

Given the probability distribution function: x 0 ...

Key Concepts

Mean (Expected Value)

The mean, or expected value, is a measure of the central tendency of a random variable. It is computed by summing the products of each value and its corresponding probability. This value represents the long-run average outcome of experiments or trials when the random variable is observed repeatedly.

Variance

Variance quantifies the spread or variability of a random variable around its mean. It is calculated as the expected value of the squared deviation from the mean. A higher variance indicates that the values are more spread out, while a lower variance suggests they are closer to the mean.

Probability Distribution Function

A probability distribution function (often called a probability mass function when dealing with discrete variables) describes the likelihood that a discrete random variable takes on each of its possible values. It assigns a probability to each value in the sample space, and these probabilities must be non-negative and sum to one.

Cumulative Distribution Function

The cumulative distribution function (CDF) provides the probability that the random variable is less than or equal to a given value. It is obtained by summing up the probabilities for all outcomes less than or equal to that specific value, and its graph is a non-decreasing step function for discrete random variables.

Transcript

0:00 Welcome to numerate.

00:02 In the current problem we are given the random variable x with the probability distribution we will write it as px with three values 0 1 and 2 and the probabilities are respectively 0 .25 0 .5 and 0 .25 so if we see the total of these 3 values will all always be one and hence we are confirmed that this will be a probability distribution.

00:37 So the first task that we have here is to draw a pmf of these values.

00:44 So if we start by drawing the axis, you know, 0 ,1, and for example, say we say this is 0 .25 and this is 0 .5 and so on.

01:03 So, for x equals to 0, the probability is at 0 .25.

01:08 So, we draw a line like this.

01:11 Okay, it should be matching exactly at this point, but because i'm drawing free hands, so i hope you understand that.

01:17 Again, for x equals to 1, this should be the point, correct? so we draw again a straight line like this.

01:24 Clearly, it's not a straight line.

01:26 And then again, like, oops, this is a little bit.

01:32 Yeah.

01:33 So, this will be the answer to the first problem.

01:37 Now, the second problem says find out the cumulative distribution.

01:42 So, if it is x and here it is 0, it is 1 and over here it is 2.

01:52 And probability x less than equals to x will be, here it will be 0 .25.

02:02 Why? because probability x less than equals to x will be 0 .25.

02:03 Why? because probability x less than.

02:04 Than equals to 0 if i write it like that x less than equals to 0 is nothing but p 0 but when you think probability of x less than equal to 1 it will be nothing but probability x equals to 0 and x equals to 1 so it is p 0 plus p 1 so we get 0 .5.

02:30 Proceeding with the same logic probability x less than equals to 2 that will give us p 0 plus p 1 plus p 2 so we will get 1 so now if we plot this in a graph what would it look like so we have 0 we have 1 we have 2 correct so suppose this is 0 .25 this is 0 .5 this is 0 .75 this is 0 .75 this is 0 .75 this is 0 .75 and probably here 1 .0 so the first this is okay let me just clearly tell you this is 0 .25 and this is 0 .5 right so we should be having here 0 .75 i'm sorry for the little bit confusion for you there so now if we see for 0 it is 0 .25 right so we have a point over here exactly like the previous diagram but we will see this there is an upward increasing trend.

03:36 The values will keep on increasing.

03:38 So the second value for x equals to 1, it is 0 .75.

03:42 So it will be here.

03:44 Correct? so you see there is an upward pattern.

03:48 Here also there was an upward pattern but then it could decrease, right? it could decrease.

03:52 But in cdf, cmf, we will say it cannot increase upwards.

03:59 Correct? so i'm sorry, in cmf it will always increase it will not go downwards like it can happen in regular probability distribution so what will happen it is at 1 so if we draw a line like this you see there is an upward trend so it will always keep on going up okay that's why because it is cumulative probability it is always getting added so either it will increase okay or it will stay at that level right after this if you think what is the probability of x less than equals to 3 or probability x less than equals to 100 it will be still be the same it's one so it will always be one one one for each of the upcoming points but you don't want this part for your answer the answer is only this part okay now so this is part b that i solved this and this now for this next part we have to find the mean.

05:02 Now what is the mean of a variable? it is value into probability for all the values, correct? so, what is the first value? 0 into what is the first probability 0 .25 plus 1 into 0 .5 plus 2 into 0 .25...

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