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in this video we're going to focus on uniform distribution probability problems and so we have an example problem to start right here number one the amount of time a person must wait for train to arrive in a certain town is uniformly distributed between 0 and 40 minutes part a determine the probability density function f of X so if you know what to do and you want to try it feel free go ahead pause the video and work on this example now what you need to understand is that if you have a continuous probability distribution the area under the curve that is the total area must equal to 1 and also if this distribution if it's uniformly distributed that means that f of X is constant it has a constant value over any equivalent interval of values between 0 and 40 in the case of this example but now how do we determine f of X well let's draw a graph now f of X is going to be constant as we said before because it's uniformly distributed so this is gonna have a horizontal line and this is going to be from the interval A to B so this is our f of X value or our y value in this graph and we said that the area under the curve that is the area of this shaded region which is basically the area of a rectangle that's equal to 1 now what is the area of a rectangle from geometry you know that the area of a rectangle is equal to the base times the height so this is the base let's use capital B to distinguish it from a lowercase B so the base is basically as you can see it's B minus a now what about the height what's the height the height of the rectangle is right here notice that it's equivalent to f of X so let's replace H with f of X and the area for any continuous probability distribution problem is 1 so now let's solve for this variable f of X to do so we needs to divide both sides by B minus a and so the probability density function f of X is simply 1 over B minus a so that's the formula that we're going to use to get the answer for Part A now in order to finish Part A we need to determine what the values for a and B are in this example so we were told that the time that the person must wait for the train to arrive in a certain town is going to be between 0 and 40 minutes so a is the minimum value a is 0 B is the maximum value which is 40 so now we can calculate f of X it's going to be 1 over 40 minus 0 or simply 1 over 40 so as we said before for a uniform distribution f of X is constant it's simply equal to a number with no variables attached now what about Part B how can we draw a graph of f of X I'm gonna put this graph on the right side so here's the y value this is going to be 1 over 40 and it's constant so we're gonna draw a horizontal line a is 0 B is 40 and so that is our graph for f of X it's gonna look like a rectangle that's all you need to do for Part B now sometimes you may get asked to write the constraint values for X if you ever get a question like that just know that X is always between a and B so in this example X is between 0 and 40 but now let's move on to Part C what is the probability that a person must wait less than 8 minutes what do you think the answer is to that problem so what is the probability that X is less than 8 now what you want to do is represent this on a graph so 8 let's say it's somewhere over there we need to find the area to the left of this line so here is the shaded region what is the area of that rectangle so the base as we can see is 8 the height is not going to change that's 1 over 40 so area is base times height let's use capital B so B is 8 H is 1 over 40 so this gives us 8 over 40 now 8 over 40 what is that well we know that 40 divided by 8 is 5 so 8 over 40 has to be 1 over 5 now 1 over 5 1/5 is point 2 or point 20 as a decimal Oh point 20 is 20% so what this means is that there's a 20% chance that the person must wait for 8 or less minutes for the train to arrive now let's move on to Part D what is the probability that a person must wait more than 30 minutes so based on the previous example go ahead and try this one so let's say 30 is right here so we need to calculate the area to the right of 30 or between 30 and 40 so notice that the base of that rectangle it's 40 minus 30 which is 10 and the height is still 1 over 40 so the probability that X is greater than 30 or you could say the probability that X is between 30 and 40 because these two will have the same answer it's going to be equal to the area of that rectangle so it's a base times height base is 10 height is 1 over 40 so this becomes 10 over 40 if you cancel a 0 that's 1 over 4 1/4 as a decimal is point 2 5 if you multiply that by 100 you get 25% so there's a 25% chance that the person must wait more than 30 minutes for the train to arrive now let's move on to Part II so calculate the probability that X is between 10 and 26 or the probability that a person must wait between 10 and 26 minutes for the train to arrive so what we're gonna do is we're gonna create a new graph since that one has been used so extensively so here's our rectangle from zero to 40 f of X is still the same now we want to calculate the area between ten and twenty six so what is the area of that rectangle so once again area is equal to the base times the height and as we can see the base is between ten and twenty six so it's 26 minus ten and the height is still one over forty now 26 minus ten that's going to be 16 and 16 times one over forty is simply 16 over 40 now we could simplify that fraction 16 is 8 times 240 is 8 times five 8 divided by 8 is 1 and so the area is 2 over 5 now 2/5 as a decimal if you type that into your calculator that's gonna be point 40 so the probability that X is between ten and twenty six is going to be point forty or forty percent now what about the next one what is the probability that X is equal to 20 I'm gonna give you a minute to work on that one go ahead and try it what do you think the answer is now if you're not sure sometimes it's best to draw a graph so here's 40 here's 0 now notice that we don't have a range of values we simply have one x value so therefore we have a line at x equals 20 notice that there is no rectangle for line therefore the area is going to be 0 if you try to use the formula base times height the base is 0 the base is usually the difference between two values but you only have one value so there's no way you can calculate the area for that it's going to be 0 now what about the next one where X is greater than 45 by the way going back to this problem anytime you see a problem where X is equal to a number let's say if X is equal to 35 or 15 it's always gonna be 0 unless you have an interval values or two different X values like this one if you don't have it it's going to be 0 now the probability that X is greater than 45 if we were to extend this graph 45 would be somewhere here but notice that f of X is 0 beyond 40 so if you were to try to use the formula base times height your base could extend to infinity but your height is zero so basically you have no rectangle so if the height is zero the area is going to be 0 therefore the probability that X is greater than 45 because it's outside of the range of 0 to 40 the probability is going to be 0 so that's the answer for that one now let's move on to Part F calculate the mean and the standard deviation so for uniform distribution the mean is going to be a plus B divided by two it's basically the average of the minimum and the maximum the mean and the median for a uniform distribution is the same because it's symmetric on in to the left side and the right side of the mean a is zero be the maximum value is 40 so the average of zero and 40 is going to be 20 so that is the mean for this problem now let's calculate the standard deviation here's the formula for the standard deviation it's going to be B minus a divided by the square root of 12 so that's going to be 40 minus zero divided by the square root of 12 40 divided by the square root of 12 is eleven point five four seven so you can round that to eleven point five if you want to so that's how you can calculate the standard deviation for this problem now what is the 85th percentile feel free to pause the video and try it if you want to so let's go back to our graph now let's break this up into four equal parts the mean is 20 the minimum zero is basically the zero percentile the mean is at the 50th percentile and 40 which is the maximum that's at the hundredth percentile ten is at the twenty-fifth percentile and thirty is at the 75th percentile so the 85th percentile is somewhere between 30 and 40 let's call that value K so k is the value that we're looking for that corresponds to the 85th percentile how can we find K in this problem first you need to understand that the 85th percentile corresponds to the area to the left I said that wrong to the left rather of the K line so the area of this rectangle that we shaded is 85% or 0.85 so the area on the left side is going to be equal to the base times the height and the area is the same as the probability for any continuous probability distribution function so we can replace the area with 0.85 now the base of the rectangle is the difference between K and 0 so it's K minus 0 the height is still the same that's 1 over 40 so we have 0.85 is equal to K minus 0 is simply K and then times 1 over 40 to get rid of the fraction we can multiply both sides by 40 so the 40s will cancel on the right side and as K is simply 40 times 0.85 so because this particular problem was centered at zero or it started at zero the 85th percentile is simply 85% of 40 40 times 0.85 is 34 now is this answer reasonable does it make sense 34 is between 30 and 40 so that's a reasonable answer so that's how you can calculate the value of a percentile given a uniform distribution problem now for the sake of practice let's try another example you could work on it if you want to go ahead and pause the video and you know give it a shot number 2 the amount of time that it takes a student to complete a chemistry test is uniformly distributed between 20 and 45 minutes part a write the probability density function f of X so the probability density function or the PDF is going to be 1 over B minus a so now what are the values of a and B in this example but we could see that X is between 20 and 45 so therefore a is 20 in this example and B is 45 so this becomes 1 over 45 minus 20 45 minus 20 is 25 and so f of X is equal to this now let's move on to Part B draw a graph now a is 20 and B is 45 and let's say this is f of X which is 1 over 25 all I need to do is draw a rectangle between 20 and 45 so that's it for Part B now for Part C what is the probability that a student will take more than 36 minutes to complete the exam so what is the probability that X is greater than 36 so let's say 36 is right here what we need to do is calculate the area of this shaded region so we know the area is going to be the base times the height so this is the base that's the difference between 45 and 36 and then the height of the triangle it's gonna be f of X which is 1 over 25 now 45 minus 36 is 9 so this becomes 9 over 25 9 divided by 25 is 0.36 as a decimal which is 36% if you multiply by 100 so that's the probability that X is greater than 36 so that's it for Part C now let's move on to the next part Part D what is the probability that the student will take between 26 and 35 minutes to complete the test so go ahead and try that so let's say 26 is here and let's say this is 35 so I'm gonna highlight the area in red to distinguish it from the last one so this right here is the new base so the area of the triangle I mean another triangle but the area of the rectangle shaded in red is gonna be the base times the height so the base is 35 minus 26 and the height is still the same now 35 minus 26 is still 9 so the answer is gonna be the same as last time that is going to be 0.36 so the probability that X is between 26 and 35 is once again 36% now let's move on to Part II so first let's clear away some stuff let's start with the median now because we're dealing with a uniform distribution the mean and the median are the same so we could use this formula a plus B divided by two so that's gonna be twenty plus forty five divided by two so that's 65 divided by two sixty-five is basically 64 plus one so we can write it like this half of 64 six divided by two is three four divided by two is two so 64 divided by two is 32 plus a half so it's gonna be thirty two point five so that is the mean now let's calculate the next thing the variance so the variance is gonna be B minus a squared divided by twelve B is 45 a is 20 so 45 minus 20 is 25 25 squared is 625 and 625 divided by 12 is 52 point zero eight three so that's the variance now the standard deviation is simply the square root of the variance so it's going to be the square root of 52 2008 three and so the standard deviation is seven point two one seven so that's it for Part II now let's move on to Part F what is the value of the third quartile to get the answer for this one it's gonna be helpful to draw a graph so this is a this is B and the y value is 1 over 25 now we're gonna break this interval into four sub intervals we said the median was thirty two point five the median is the second quartile this is the first quartile and this is the third quartile the first quartile is the 25th percentile the second quartile is the 50th percentile and that third quartile is the 75th percentile so what we need to do is determine the value that corresponds to the 75th percentile which is the third quartile and we're going to call it K so the area of the rectangle to the left of K is 0.75 because it corresponds to the 75th percentile so the area of that rectangle on the left is going to be the base times the height so the area is point seven five the base the left of the base of the rectangle is the difference between K and 20 so we're gonna say K minus 20 and as we know the height is f of X the probability density function which is 1 over 25 so now we need to do some algebra let's begin by getting rid of this fraction by multiplying both sides by 25 so these will cancel so 25 times 0.75 is 18.75 and so we're left with this on the right side which is K minus 20 the last step is to add 20 to both sides so 20 plus 18.75 that's going to be 30 8.75 and so this right here is the 75th percentile and as we can see the answer makes sense because it's in between 30 2.5 and 45 just as 75 is between 50 and 100 if we were to average these two numbers it would give us the answer 32.5 plus 45 is seventy seven point five if you divide that by two it will give you thirty eight point seven five now let's move on to the last part part gene what is the probability that the student will take more than 40 minutes to complete the test given that he always takes more than 30 minutes to complete any chemistry tests so we have the keyword given go ahead and try this problem so first we need to recognize that we need to calculate a conditional probability the formula for conditional probability let's say if we want to find the probability that event a occurs given that event B has already occurred this is the probability of getting a and B this is a the intersection between a and B divided by the probability of event B occurring so in terms of this problem we want to find the probability that X is greater than 40 given that X is already greater than 30 so it turns out that this is equal to the probability that X is greater than 40 divided by the probability that X is greater than 30 and let's talk about why why that's the case now looking at the first formula we can illustrate this with a Venn diagram let's say a is on the Left B is on the right so where the two circles overlap that is the intersection of a and B so the probability that a will occur given that B has already occurred is basically the area of this portion that is the intersection of a and B divided by the entire area of the circle that's represented by B now in this problem we're gonna do something similar so let's begin with a picture so let's say this is 20 this is 45 and here is our rectangle in this case B would be the area from 30 to 45 so 30 would be here so I'm gonna highlight B in red now a would represent the area that is between 40 and 45 so a would be this region so notice that the region where a and B overlaps is between 40 and 45 because you have both a and B shaded in that region and then the region the entire region of B would be from 30 to 45 so that's why this formula is going to work so now the probability that X is greater than 40 that's going to be the area of the rectangle highlighted in blue so the base of that rectangle is 45 minus 40 times the height which is 1 over 25 now for this one the probability where X is greater than 30 the base of that rectangle is from 30 to 45 and the height is the same so we could cancel 1 over 25 45 minus 40 is 5 45 minus 30 is 15 so 5 over 15 15 is basically 5 times 3 5 is 5 times 1 we can cancel a 5 and get the answer 1 over 3 and 1 over 3 as a decimals basically point three repeating so the probability that the student will take more than 40 minutes to complete the test given that he always takes more than 30 minutes to complete any tests that answer which I'm gonna write here is let's multiply this by 100 so that's gonna be 33.3% with the 3 repeating so that's it for this video now you know how to solve probability problems related to uniform distribution situations thanks again for watching