Find the Probability Density Function for Continuous Distribution of Random Variable

ha monokuma will discuss continuous probability distribution where we have modeled the distribution with the equation the question here is construct the function f of X equals to K minus x over 4 for X between 1 and 3 and equals to zero otherwise which is being used as a probability density function for a continuous random variable capital X right so these X are the values for the variable capital X okay so that is how the question is you need to find the value of K and second part of this question is find probability that X the random variable is less than equal to 2.5 let us see how to solve such a question now as given to us that the function f of X equals to K minus x over 4 when X is greater than equal to 1 and less than equals to 3 is a probability distribution probability density function right that's given to us for a continuous random variable now what does that mean now if f of X is a probability density function in that case the area under this curve within the given region should be equal to 1 right so area we can find with integration so integration of f of X within the limits 1 to 3 should be equal to 1 that is by definition for probability density function correct so we'll do integration of this which is from 1 to 3 the function given to us is K minus x over 4 D of X and this is equal to K X so within the interval it is equals to K X there is a constant minus x square over 2 and this is four already right that is what it is well in the interval 1 to 3 so that is what we can okay now this should be equal to one is there okay so this should be equal to one so we are solving this equation where this integral should be equal to one so let us simplify this part then we write one equals two and here when we write three here we get 3x my I'm sorry 3k three times K minus three is nine nine over eight so this is when we substitute three in this equation takeaway when I substitute one I get K minus one over eight okay so what we get here is three K minus K is 2 K and minus nine over eight K this becomes plus one over eight K is is eight over eight K right which is minus one now when you simplify this this implies that two K equals to take this one on the other side equal to 2 or K equals to two divided by two which is what is okay so we find the value of KS one so we get the sunset k equals to 1 is it ok so that is how you are going to find the value of unknown constant where K is a constant in radom where K is a constant now we know what function is right now it says find the probability when X is less than equal to 2.5 so that probability will be sum of all the probabilities within this given interval and this interval is valid from 1 remember that right it is 0 otherwise it is only valid from 1 onwards so let's try to sketch and understand this function also when we find K equals to 1 in that case what does this function look like ok let me sketch a graph to show you that part so let's say these are the x-values this is the function f of X and now f of X is equals to K is 1 for us 1 minus x over 4 right and this is 0 otherwise so this is let's say this is 1 for us ok this is 1 for us that is 2 and let's see this is 3 for us so it is 0 otherwise so 0 otherwise but from 1 to 3 the value is 1 minus x over 4 correct so 1 minus x over 4 really means that is a straight line with y-intercept of 1 but it's just kind of dropping down kind of kind of going down like this straight line dropping down right so we can find the value of the function at 1 let's do that let's sketch this part it becomes simpler so what is f of 1 equals 2 when I write 1 for X I get 1 minus 1 over 4 right so which is 3 over 4 correct so 304 and what is therefore 3 equals 2 so when I write 3 here we get 1 minus 3 over 4 so that is 1/4 correct so let us say this is 3 over 4 for us so add 1 we have a point 3 over 4 and let's say this is 1 over 4 so at 3 we have this point so the line which represents the probability density function is kind of like this is that ok where these points are three over four and one over four correct and what we just found was the idea between all this is one right half base into height base is 3 minus 1 which is 2 and height is 3 or 4 minus 1 over 4 which is half is it'll be 2 over 4 when you multiply the area of 1 and that shows that it is indeed a probability density function since the area under this curve is 1 correct so that is how you actually solve these questions so area under the curve is 1 now the question is find the probability when X is less than equals to 2 point 5 that is to say we are considering a point here midway between twice so kind of here less than means up to this place is it okay up to this between this find the probability when it is between this so you can calculate the area directly from the graph or well do the integral once again so we have the integral this time let me just divide this page so you drink part 2 now which is integral of the function which now we know K is 1 so we can write this as 1 minus x over 4 between 1 & 3 1 & 3 okay so that is what we are going to do so as we did earlier this is equal to this is a constant so integral is X X integral is X square over 2 so 2 times 4 and we will find sorry we are doing between sorry between 1 to 2.5 between 1 to 2 point 5 right this value 2 point 5 okay so between 1 to 2 point 5 we can find this value is it okay that is to say we'll calculate the value for 1 minus 1 over 8 take away I mean this should have been the negative 12.5 - okay let me just change this to point five minus two point five squared okay take away one minus one over eight is okay so this what we calculate so we have two point five take away two point five square divided by 8 that is equal to 52 over 32 so this is 55 over 32 take away the best portion is 8 minus 1 which is 7 right take away 7 over 8 is it okay so we'll do this take away within bracket 7/8 and that is equal to 27 over 32 so we get our answer as 27 over 32 exactly so that is the probability for the random variable to be less than 2.5 okay so the assess 27 over 32 so we get this answer as equals to 27 over 32 I hope that helps so whenever you have a continuous probability distribution we are trying to find the area of the probability density function within that interval so that is the whole concept so area could be fine by doing the integration within the interval that should give us the area I hope that helps thank you and all the best

Channel: Anil Kumar

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Id: sBy0w_j6mgE

Length: 9min 53sec (593 seconds)

Published: Sat Dec 03 2016