 Dear students, I'm now going to discuss with you the concept of a continuous random variable. Continuous random variable is quite different from a discrete random variable. Discrete random variable, amtor pe, hum yeh count kar rahe hotein cheezin, uswag thum encounter kar rahe hotein. For example, the number of heads that I am going to get if I toss a coin four times, I may get no head, one head, two heads, three heads or four heads or har ekke against there is a probability or uske baad add karke we get one. Continuous random variable may the situation is very different. In the continuous case, we cannot define probability on one particular point. Instead, it is probability defined on an interval. Continuous random variable woh hai jo ke hum thum encounter karthein when we are measuring things. Simplest example agar aap kisi student ki height measure karhein. It depends on the refinement of the measuring instrument. Siya aap uski height kya measure karein ke. Agar hum kahein ke ji this person is five feet four inches tall. So in other words, we are saying that he is or she is 64 inches tall. So this is, you know, the measurement at a crude level, yani hum ne usme decimal bilkuli involve ne kya aur hum ne usko round karke hum ne kya diya ke ji 64 inches uski height. In reality, it is some value between 63.5 and 64.5 agar hum zara zyada refined instrument le aayin. So fir hum measure karein uski height aur hum kaheen ke it is actually a height somewhere between 64.15 and 64.25 inches. Ke nahi hum aksad hain ke aap kaya jo aap ka jo measuring instrument hain, woh kithne precision tak aapko measure karke desh sakta hai. We can have maybe five, ten, fifteen decimal places after the decimal. Lekin exact accurate hum nahi karpainge because theoretically, there can be an infinite number of decimal places after the decimal. Baharhal yeh bunyadi concept hai continuity ka jo loves hai continuous random variable. Yeh tis hai ke x axis kyu par that variable can assume any value. Aap iske baad agli baat yeh ke how do we define the probability distribution of a continuous random variable. My dear students, in this regard, let me first and foremost talk about the cumulative distribution function, the CDF, which is denoted by capital F of x. And as you can now see on the screen, capital F of x is defined as the integral from minus infinity up to x of small f of x. Aap yeh jo small f of x hai that is some function that is going to give us that algebraic expression that will be our probability distribution. Yeh jo integral hai, this is the cumulative distribution function. This gives us the probability of our random variable x falling somewhere between minus infinity and some particular value x. Iske baad, if we take the derivative, the derivative of capital F of x, that gives us small f of x, small f of x is called the probability density function. It is that mathematical equation which defines our probability distribution. What are the basic properties of a probability density function? They are two very, very important properties. The first one, that the integral of this small f of x from minus infinity up to infinity is equal to one. And the other one, that this f of x will never be negative. Yeh bohi baali baad hai, ke jo discrete ke case mein hum dosri tarah se bhiyaan karte hai. The sum of the probabilities has to be one. Yeh hape hum kate hai, ke jo integral, yani, doswe lafzo mein, the area under the curve of that function, that has to be equal to one. Aur dosri baad jo ke discrete mein bhi hum kate hai, ke probability ke never be negative. Ohi baad yeh hape is tari ke se kahthe hain ke f of x will always be greater than or equal to zero. It cannot be negative. To yeh sara jo mein aap ke saamne rakha, iska jist yeh hai, ke continuous variable ke case mein, the area under the curve of that function gives you the probability. Agar aap interested hain in computing the probability that your capital X, your n variable X lies between some two numbers a and b, then all you have to do is to integrate your small f of x from small a to small b. Isi ko hum isra bhi kaisakthe hain ke it is capital F of b minus capital F of a. To ke agar hum cumulative distribution function ke isaap se baad karein, to it is the capital F of that bigger number b minus the capital F of the smaller number a. Lekin agar hum small f of x ke terms mein baad karein, then it is the integral from small a to small b. Ab mein aap ke saamne ek simple example rakthi hum of a continuous random variable. Suppose that we randomly select a number between zero and one. Yani, it can be any decimal number between zero and one and we are assuming that it is being selected randomly. In this case, my dear students, small f of x will be written as you now see on the screen. Small f of x will be equal to one for all those x values which lie between zero and one. And small f of x will be equal to zero elsewhere. Ab yeh jo bhe ne kaha isko agar hum graphically represent karna ja hain. So, what will we get? As you can now see on the screen, we will get something like a rectangle. Niche x axis ke upar zero se one tak jo area hain jo uska interval hain. Uske directly upar level one peh we have this horizontal line. Kyuke randomly jab hum chun rahe hain to phir hum probability kisi bhi number ke chune jaane ki jo hain, that is the same. So, we get this horizontal line and if we join the upper line with the lower one to a vertical line, we get something like a rectangle. So, this is called the uniform distribution. And it is also called the rectangular distribution. Yeh contineus variable. Un mese jo bo one of the very first that we would discuss when introducing the topic anywhere, that would be this one, the simple uniform distribution where x goes from zero to one. And the height of that rectangle is also one. Baat is ke andar yeh ke jo area under the curve hain, that has to be one. Ab yeh chun ke ek rectangle hain, ish me to hum geometrically, elementary geometry kiru se bhi area dekh sakte hain. So, dekh lein ke zero to one jo niche interval hain, jo base hain, usko jab aap uski height one se multiply keringe. So, height into base one into one is equal to one. So, usrektangle ka area, yehni uske upper line ke niche jo area hain, wo baaki one hain. So, therefore it is fulfilling one of the two properties, those two very important properties of continuous probability distribution, that the area under the curve is one. And the other one, that f of x cannot be negative. So, wo jo humari line hain, that is not below the x-axis, it is above the x-axis. So, that condition is also being fulfilled. This is the concept of continuous random variable and its probability distribution.