 Dear students, let me present to you another application of Shebyshev's inequality. Shebyshev inequality ka jo ek way hai likhne ka, woh zyada well-known woh hi wala hai ke jisbe hum yeh kate hain ke, the probability of the modulus of x minus mu being less than or equal to k sigma, this probability is greater than or equal to 1 minus 1 over k square, ya usi ka dusra version. Ike in is waqt, I would like to present to you an alternative form of the Shebyshev's inequality, or woh it is not at all difficult to achieve that form, you simply put k sigma equal to m and you get this new form, so let me read it for you. If we put k sigma in that earlier version equal to m, then the inequality becomes the probability of the absolute value of x minus mu being greater than or equal to m, this probability is less than or equal to sigma square over m square. For all values of m that are greater than zero. Acha yeh jo abhi wane kaha isko hum ek aur tafa bhi kailate hain, sigma square ki jaga I will use the word variance of x or mu ki jaga I will say mean. So then what do we say? We say that the probability of the absolute value of x minus mu, x minus the mean, this absolute value being greater than or equal to m, this particular probability is going to be less than or equal to the variance of x divided by m square. So agar iss version se hum chalein, to ek example ke zaree mein show kartee hum, ke kitne interesting sahiyeh sahiyeh situation banti yeh. The example is as follows, suppose that we have random variable capital X such that the expected value of x is equal to 3 and the expected value of x square is equal to 13 and we are required to utilize the Chebyshev's inequality in order to obtain a lower bound for the probability that X lies between minus 2 and 8. I say it again because we only know that it is a random variable but we don't know what its distribution is. We cannot say that it is the normal distribution so we can apply those percentages which apply for the normal distribution. Yeh humi nahi pata ki isski distribution ki form kya hai. The only information that we have is about the expected value of x and the expected value of x square. To phir zaheer hai ki iss case mein to hum Chebyshev inequality hi sirf istimal kar sattein in order to obtain a lower bound yeh nahi exact probability nahi mil sattein. Lekin ek lower bound hume mil jaye ke usse aur kam nahi ho sattein the probability of X lying between minus 2 and plus 8. So henna interesting question is quite an interesting question so let us see how we might solve it. Alright let us look at the Chebyshev inequality once again. So we have the probability of the absolute value of X minus mu being greater than or equal to M. This probability is less than or equal to variance of X over M square for all M greater than 0. However in order to obtain a lower bound for that probability that we are wanting to deal with we will need to flip the inequalities and what will we get then when we flip the inequalities. We will read it as follows the probability of the absolute value of X minus mu being less than M. This probability is greater than 1 minus the variance of X over M square. Abhis ke baad students jo uski left hand side pey yeh absolute value jo hai uske saat pehle deal kar leh. Kyuke wo aapko thuri confuse kar rahi hoge aur yeh bada important hai ke aap uske saat deal kar nahi jaan leh. When I say the absolute value of X minus mu is less than M I can rewrite this as follows minus M less than X minus mu less than M. Yani, dosre lafzo mein hum yeh kere hain yeh jo X minus mu, yeh jo entity hain it lies between minus M and plus M. To yeh jo hoge hain iske baad yeh step aur bhi hain. Yeh jo X minus mu beech mein likkav hain iska jo mu jo hain usko hum wahan se hatana chaat hain aur hum chaat hain ke sirf X hoz jaga pe. So, I will take mu to the left side of that inequality and also to the right side of that inequality. So, what will I get then? I will get mu minus M less than X less than mu plus M. Kyuke zahir hai ke jo minus mu tha beech mein usko aap idhar aur udhar jab leh jayenge to wo plus ho jayenge. Now that this is clear, we are able to proceed further. The form that I now have is the probability of mu minus M being less than X being less than mu plus M. This probability is greater than one minus variance of X over M square. Aap variance of X to mera khala pehle hum compute karlein Kyuke jab usme wo chahiye humko, to we must compute that. Aur duke hain hain hain hain usse pehle hi expected value of X square aur expected value of X available hain. So, then we can apply the shortcut formula and we can get it very quickly. So, students, what is the variance of X? E of X square desa ke thori der pehle aapko bataya that was equal to 13. Aur E of X jata that was 3. So, 13 minus 3 square, 13 minus 9, that is equal to 4. Variance ab hain hain hain hain hain hain hain hain to wo hum shabi-shab inequality apply kar sakenge. Lekin abhi thoda sa tricky part rahta hain ke wo humhe kyaise pata chalega ke M kya hain pehle chalee zara trial and error se isko So suppose that we put M equal to 5. If we keep M as 5, then mu minus M that will be 3 minus 5 and that will be equal to minus 2. Similarly, mu plus M that will be 3 plus 5 and that will be equal to 8. So you have seen that your Shebyshev inequality, the final version which we are dealing with right now, the left hand side of that is equal to exactly what we wanted to compute the bound for. So what is that? Probability of minus 2 less than x less than 8. That is the probability of x lying between minus 2 and 8. So now that we have identified the value of M that is 5, then we put Shebyshev inequality. So what do we have? We have the probability of minus 2 less than x less than 8 according to Shebyshev inequality that is going to be greater than 1 minus the variance over M squared meaning 4 over M squared, meaning 4 over 25. So 1 minus 4 over 25 is 21 over 25 or in other words 0.84. So we have been able to determine the lower bound for this particular probability. Interpretation is the distribution of which we do not know the form. But this information is available that expected value of x is equal to 3 and the expected value of x squared is equal to 13. For this particular distribution the probability that x will lie within minus 2 and plus 8 this probability is going to be greater than 0.84. Meaning 0.84 is more than or less than or less than. So we have obtained my dear students the lower bound for this probability.