 स disproportionately generating function ntycap students once you have the basic concept of the probability generating function and its utilization let me give you the algebraic expressions of some well-known bgfs ती हैखीं सी टीा है के जिस टीा आप आप एक सप्ट करते हैं के, जो वैल नोन डिसम्बूचींस हैं, उनके वोम имеण जं़रेर्टिंग फुक्षृनच के एक श्फ्रशंस हमे याद होने चाहीें, तो आप सा क्या आप आप छले हैं। because they are so well-known. इसी तरा, same is the case for the PGF. So let me put in front of you the expressions of some of the well-known ones. Let me begin by the PGF of the constant random variable. What I mean is that I am talking about that case when it is actually not a variable but a constant. जिस्कोहाम दीजेनरेट दिस्रिवूशन भी के तें. तो इस दिस्रिवूशन is defined on just one single point then the PGF will be given by जी रेज तो सी where C is that particular point where the entire probability lies. इसके बाद let's talk about the PGF of the Banuli random variable. अगर फेर कोईन हो, तो फिर half plus C by 2 answer आता है. लेकिन अगर फेर नहो. इस the probability of getting a head is P and the probability of getting a tail is Q then what is the expression of the PGF of the Banuli random variable X. Well it is very simple. It is Q plus PZ. अगर Q and P तोनो half है. तो जाहे रहे के वो ही आजाएगा half plus Z by 2. अगर in general it is Q plus PZ. इसी तरा जो geometry distribution है उसका जो PGF है, उसका expression PZ Divided by 1 minus QZ जब हम derive करते है, तो this is what we obtain. और उसके अंदर एक condition भी है, अवो ये है के the modulus of Z has to be less than 1 over Q. इसके बाद बिनोमिल random variable की बाद करते है, which is the most well known random variable in the discrete situation. इसका जो probability generating function है that is given by Q plus PZ whole raise to N. अचे देखे आप को पता तो है के Bernoulli is a special case of the binomial. अई मींद के अगर आप binomial का जो parameter है N, number of trials अगर उसको 1 रग दें, वो बरनूली वाला case बन जाता है. तो आभी जो में आपके सामने रखा है expression PGF का बीनूल रान्दम वेरिबल के लिए Q plus PZ whole raise to N. अब अगर उसके शो लए खन को 1 रख दें, तो क्या में लेगा? वी थे पड़ा षलता ऐब ने अप एक मिने भाज ताल से बता है, के जो बरनूली रान्टम वेरिबल है उसका जो होता है तो यहाँ भी आनको वन रखें और वही आजाता है, योंके उनका अपिस में रिलेशिनची पी आई है, इसके बाद पोईसोर आन वेरिबल की बाद कर लिजे, again it's a very well-known distribution and the pgf is given by e raised to lambda into z-1. For the negative binomial distribution, my dear students, the pgf is given by pz over 1-qz whole raised to n and this may be the same condition that modulus of z has to be less than 1 over q and of course it is understood that the p and q that we are saying is p plus q is equal to 1. Okay, I have kept this in front of you. A very well-known distribution, one of the very first distributions that comes to mind in the case of the discrete variable, that is the discrete uniform distribution. So this pgf, let's try to take it out ourselves. We should also remember that this is the pgf because it's a very well-known distribution. Let me take the example of the 10 digits 0 to 9. 0, 1, 2, till 9, obviously there are 10 digits and if we randomly select any digit, that is the perfect example of a discrete uniform distribution. So as you can now see on the screen, we can represent it as the two columns x going from 0 to 9 and p of x being 1 by 10, 1 by 10, 1 by 10 for each and every one of those x values. Since all these probabilities are equal, that is why it is called a uniform distribution. Now if I want to take this out, probability generating function, then I will construct the column of z raised to x. So z raised to 0, that is 1, z raised to 1 is z, z squared, z raised to 3 and the last one in this case, that will be z raised to 9. Now I will multiply the probabilities corresponding to these columns so that I get the final column, the sum of which I take, that gives me the probability generating function. So when I do that, you can easily see that I will obtain p of x into z raised to x as 1 by 10, z by 10, z squared by 10, z cubed by 10 and so on. Now what we have to add to these values, these products, so you can see that 1 by 10 is common in each one. So we will take out 1 by 10 and we will get the expression 1 plus z plus z squared plus z cubed, so on, so on plus z raised to 9. So you are seeing that this is not an infinite but a finite geometric series. It is a simple example of a geometric series and in geometric series you know that the first term is called a which is 1 in this case and the common ratio is called r which in this case is z because if you divide any term from the previous term then you will get z as the common ratio. n is the number of terms, students, it is not 9 as you can see that number is 10. So all I have to do now is to apply the formula of the sum of a finite geometric series so as you know school level we have done these things. This particular sum is given by a into 1 minus r raised to n whole divided by 1 minus r so therefore in this case it would be 1 into 1 minus z raised to 10 divided by 1 minus z and 1 by 10 you remember we also removed the common so we also attached it to it and that is then the probability generating function of this particular discrete uniform distribution exactly what is it g of z is equal to 1 by 10 1 minus z raised to 10 over 1 minus z