 Dear students, I'm going to present to you the concept of a random vector. Random variable, you know, you see what is a random vector. So I will do it with reference to an example. Suppose that a coin is tasked three times and our interest is in the ordered pair. Number of heads on the first two tosses. That's the first part of my ordered pair and number of heads on all three tosses, that is the second part of my ordered pair. Look, usually we say that the ordered pair X, Y, that's the first part, Y, that's the second part. Here I am defining that the first head that is the number of heads on the first two tosses and the second head that is the number of heads on all three tosses. So now you have to think about what is possible. If we let C denote the sample space, let us see what are the various possibilities, all possible outcomes that I could have if I am tossing three coins together. So obviously we can start from the T side or from the H side, it's the same thing. So let's start from the tail side. Tail, tail, tail, tail, tail, head, tail, head, tail, tail, head, head, tail, head, tail, head, head, head, head, head, head, tail, and last but not the least, head, head, head. So sample space, you know that the set of all possible outcomes are done. And it should have been R, because 2 raised to 3 is 8. Every time we toss, there are two possibilities. If we do it three times, it becomes 2 raised to 3 RT. Now, let X1 denote the number of heads on the first two tosses and X2 denote the number of heads on all three tosses. Then our interest can be represented by the ordered pair X1, X2. So now let's see for example what are the possible values of X1. If we evaluate X1 for that particular outcome, which is head, tail, head, then what will come? Students, do you want to say that it is equal to 2? That is not correct. Look, what was X1? The number of heads on the first two tosses. And we have a sequence, head, tail, head. So first two, look, head, tail, number of heads, 1. X2 is the number of heads on all three tosses. That of course is equal to 2, because head, tail, head is 2. So in this way, we can look at all the possibilities. And then what do we get eventually? We can talk about space. There's a A-T, sample space. Now we can talk about the space D, which contains the following ordered pairs. 0, 0, 0, 1, 1, 1, 2, 2, 2, and 2, 3. You may be thinking that something has been missed. Why did I not say 1, 0? And why did I not say 2, 1, etc.? So look carefully, that in this way, they are actually not possible. So what you have already told, what is written in front of you, what we have written there, which are actually possible. Let me explain this. Why did I not write 1, 0? Because if 1 is coming from the first two tosses, this means that in the first two tosses, there is one head, then how can 0 be after the comma? What was after the comma? Number of heads in all three tosses. If there is already one head in the first two tosses, then how can 0 be in all three tosses? So in this way, you can think of all of them. And D, that is in front of you, contains only those particular ordered pairs that are actually possible. Now, I want you to know that these ordered pairs, if we take this for algebraic, bracket x1, x2, bracket close, this is, this can be called a vector function from C to D. Because C, which was sample space, we didn't write 1, 2 there, there is head, tail, head and all that. So now, the way we have defined x1 and x2, because of that, we can make this statement that the ordered pair, x1, x2, is a vector function from the space C to the space D. In simple words, my dear students, aap se vithnayadratne, that the pair of random variables, x1, x2, is called a random vector. Or formal definition iski johay, let us look at that also. It is as follows from a random experiment with a sample space C, consider two random variables, x1 and x2, which assign to each element, small c of the space capital C, one and only one ordered pair of numbers, capital X1 of c equal to small x1 and capital X2 of c equal to small x2. If this is the situation, then we say that capital X1, capital X2, that ordered pair is a random vector. Ye sari baat joh main aapke saamne abhi rakhi, this was for the case when we have only two random variables. Lekin, you must note that this concept is extendable to three, four, or more random variables. Ek aur baat note kane, ke anthorpe aksar, we use the vector notation, capital X bold face denotes the random vector and it is written as being equal to the ordered pair, if it is two, the ordered pair x1, x2, lekin uske saath, ther hum lagaate hain ek prime jesa sign, which actually represents the transpose of the row vector x1, x2. Yehni matrix algebra ki roose, agar ek row vector hain, uska agar hum transpose lehte hain, so that is a column vector. To anthorpe jab hum, books wagaara mein bhi aap dekh hain or research papers wagaara mein bhi dekh hain, so we write it like this, that we write it like a row vector, but then we also apply the transpose sign or the prime in order to represent that random vector. Ek aur baat aur wo yeh hain, khar hum conversely baat karein, to hum yeh kahe sakte hain, yeh, if the ordered pair x1, x2, in case of two, is a random vector, then pooth, x1 and x2 are random variables. Yehni mera matlab yeh hain, ke donoh taraf se yeh statement banti hain, if x1 and x2 are random variables, then x1, x2, that thing is a random vector, transpose laga hain na bhi laga hain to, aur conversely, if x1, x2 ordered pair is a random vector, then x1 and x2 are random variables.