1 00:00:00,150 --> 00:00:05,190 Before we actually get to talking about probability math functions, let's lay some foundation and talk 2 00:00:05,190 --> 00:00:06,780 about functions in general. 3 00:00:06,810 --> 00:00:12,600 For instance, let's think about this graph here, which is the graph of the function. 4 00:00:12,720 --> 00:00:15,770 F of x equals x minus one. 5 00:00:15,780 --> 00:00:19,560 We want to think about functions as inputs and outputs. 6 00:00:19,560 --> 00:00:25,710 And what they tell us is that for any value we input for X on the right hand side here the function 7 00:00:25,710 --> 00:00:30,140 will return to us a value for this left hand side, this F of x value. 8 00:00:30,150 --> 00:00:36,660 So for instance, if we pick x equals one, then the function gives back to us one minus one or zero, 9 00:00:36,660 --> 00:00:45,120 which means if we set up down here a table of inputs and outputs, so we'll have our inputs X and our 10 00:00:45,120 --> 00:00:46,560 outputs F of x. 11 00:00:46,560 --> 00:00:52,860 We said that if we pick an input of x equals one, our output is one minus one or zero. 12 00:00:52,980 --> 00:00:57,090 If we pick an input of x equals two, we get two minus one or one. 13 00:00:57,090 --> 00:01:00,030 So an input of two gives an output of one. 14 00:01:00,030 --> 00:01:05,790 An input of three would give an output of two etc. And of course that makes sense because this function 15 00:01:05,790 --> 00:01:10,950 here is just telling us whatever value we put in for x, we're going to subtract one from it to get 16 00:01:10,950 --> 00:01:11,460 the output. 17 00:01:11,460 --> 00:01:17,550 And so we can see that all of the outputs are one less than the input values and we can see that in 18 00:01:17,550 --> 00:01:18,810 the graph as well. 19 00:01:18,840 --> 00:01:22,200 Obviously all of these then become coordinate points. 20 00:01:22,200 --> 00:01:26,040 So we have this coordinate .10 and we see that. 21 00:01:26,990 --> 00:01:27,800 Right here. 22 00:01:27,830 --> 00:01:30,730 We also have this coordinate .21. 23 00:01:30,740 --> 00:01:32,930 We see that point here at two. 24 00:01:33,050 --> 00:01:38,180 One, we see that it's along this line and we have the 0.32. 25 00:01:38,210 --> 00:01:44,420 So we go out to X equals three and up to two and we sketch in that point. 26 00:01:44,510 --> 00:01:51,140 So what this function is saying is that if we pick a value X equals two, we come up to the graph of 27 00:01:51,140 --> 00:01:56,630 the function and then we move straight over to the vertical axis and we see that the value we get back 28 00:01:56,630 --> 00:01:57,320 is one. 29 00:01:57,320 --> 00:02:04,640 So when we pick two, we get back a value of one, or if we pick three, we come up to the function 30 00:02:04,640 --> 00:02:06,200 at x equals three. 31 00:02:06,200 --> 00:02:11,180 And then from here we go over to the vertical axis and we find two. 32 00:02:11,180 --> 00:02:13,970 So if the input is three, the output must be two. 33 00:02:13,970 --> 00:02:16,100 If the input is two, the output must be one. 34 00:02:16,100 --> 00:02:22,100 If the input is one here, the output must be zero, so zero and one. 35 00:02:22,100 --> 00:02:27,980 So we can get these input output pairs just from the function as we showed with our table here. 36 00:02:27,980 --> 00:02:33,830 Without ever having the graph, we can also get the input output pairs from the graph itself and the 37 00:02:33,830 --> 00:02:38,000 input output pairs from the graph will always match the values we get in our table. 38 00:02:38,000 --> 00:02:44,510 The values that are generated when we plug in values of X to our function well, a probability mass 39 00:02:44,510 --> 00:02:51,950 function works the same way, except that we plug in values that the variable can take on and what's 40 00:02:51,950 --> 00:02:54,170 returned to us is a probability. 41 00:02:54,170 --> 00:02:59,720 So here in this scenario, what was returned to us was a value along the vertical axis here, a value 42 00:02:59,720 --> 00:03:02,120 for F of X in a probability mass function. 43 00:03:02,120 --> 00:03:07,760 What we get back is a probability, a probability that that input value occurs. 44 00:03:07,760 --> 00:03:12,380 So as an example, let's look at this probability mass function. 45 00:03:12,560 --> 00:03:16,820 This function models the probability of flipping heads. 46 00:03:16,820 --> 00:03:19,250 If we flip a coin two times. 47 00:03:19,250 --> 00:03:22,370 Now, we've already looked at probability with coin flips. 48 00:03:22,370 --> 00:03:29,810 So we know that when we flip a coin two times we can flip heads and then heads we can flip heads and 49 00:03:29,810 --> 00:03:30,710 then tails. 50 00:03:30,710 --> 00:03:35,330 Or we can flip tails and then heads or tails and then tails. 51 00:03:35,330 --> 00:03:40,910 There are four possible equally likely outcomes when we flip a coin two times. 52 00:03:40,910 --> 00:03:46,520 Now, if we think about the number of heads we can get over the course of two coin flips, we can either 53 00:03:46,520 --> 00:03:49,670 get two heads here, which we see in this outcome. 54 00:03:49,670 --> 00:03:53,210 We can flip heads once, once or zero times. 55 00:03:53,210 --> 00:03:59,240 So the only possible options are flipping heads zero times, one time or two times. 56 00:03:59,240 --> 00:04:00,890 And that's what we see here. 57 00:04:00,890 --> 00:04:05,630 In the probability mass function x can be zero, it can be one or it can be two. 58 00:04:05,630 --> 00:04:13,100 So this part of the function here represents all of the possible outcomes of our discrete random variable. 59 00:04:13,100 --> 00:04:18,800 Now, it's important that we say discrete random variable because we're only going to use probability 60 00:04:18,800 --> 00:04:24,140 mass functions to model discrete random variables, not continuous random variables. 61 00:04:24,140 --> 00:04:30,320 And the reason is because we have to be able to identify a specific probability for each outcome. 62 00:04:30,320 --> 00:04:36,080 And remember, with continuous random variables, we can continue to measure an infinite number of between 63 00:04:36,080 --> 00:04:36,560 values. 64 00:04:36,560 --> 00:04:42,950 So whereas in this example, where we can flip heads zero times one time or two times, if we had a 65 00:04:42,950 --> 00:04:47,990 continuous random variable, we might need to look at the probability of zero times, but then also 66 00:04:47,990 --> 00:04:56,900 a quarter of a flip or a half of a flip or 0.4782935 flips every value, an infinite number of values 67 00:04:56,900 --> 00:05:01,250 in between zero and one and an infinite number of values between zero and two. 68 00:05:01,250 --> 00:05:07,340 And we can't define a specific probability for all of those infinite number of values. 69 00:05:07,340 --> 00:05:13,460 So we only use probability mass functions to model discrete random variables where we have this countable 70 00:05:13,460 --> 00:05:17,630 set of discrete values that our random variable can take on. 71 00:05:17,630 --> 00:05:23,570 So we have to have a discrete random variable, and all of the values that we can get are defined over 72 00:05:23,570 --> 00:05:25,520 here in this part of the function. 73 00:05:25,520 --> 00:05:30,710 This part of the function gives the probability that each of these outcomes occur. 74 00:05:30,710 --> 00:05:37,730 So this part here is P of X, the probability that the specific value of X occurs. 75 00:05:37,730 --> 00:05:43,880 And we can see here with our little chart we made that of the four equally likely outcomes will flip 76 00:05:43,880 --> 00:05:48,620 heads zero times exactly one out of those four outcomes. 77 00:05:48,620 --> 00:05:54,050 So there's a one in four chance that will flip heads zero times when we flip a coin two times. 78 00:05:54,050 --> 00:05:58,070 And so a one in four chance is a 0.25 chance. 79 00:05:58,070 --> 00:05:59,480 The same thing happens here. 80 00:05:59,480 --> 00:06:05,720 The probability of flipping heads two times happens only one out of these four possible scenarios. 81 00:06:05,720 --> 00:06:09,710 So that probability is also one fourth or 0.25. 82 00:06:09,710 --> 00:06:15,440 But the probability of flipping heads exactly once happens two out of four times. 83 00:06:15,440 --> 00:06:22,400 So the probability of x equals one is two over four or one half or 0.50.5. 84 00:06:22,400 --> 00:06:26,450 So we've matched up each of these probabilities with the course. 85 00:06:26,660 --> 00:06:27,500 The outcome. 86 00:06:27,500 --> 00:06:33,620 It's important to say now to then another requirement of a probability mass function is that all of 87 00:06:33,620 --> 00:06:41,000 these P of X values here have to add to one because remember, total probability is always one or 100%. 88 00:06:41,120 --> 00:06:45,770 Every individual probability value has to fall between zero and one. 89 00:06:46,070 --> 00:06:51,320 So 0.250.5 and 0.25 all fall between zero and one. 90 00:06:51,320 --> 00:06:57,800 And these values add up to 1.25 plus 0.5 plus 0.25 is one. 91 00:06:57,800 --> 00:07:00,440 So we have the requirement that these values add to one. 92 00:07:00,440 --> 00:07:05,360 We have the requirement that we're dealing with a discrete random variable, and we just set it a little 93 00:07:05,360 --> 00:07:11,360 bit, but let's be a little bit more explicit that all of these P of X values have to be positive. 94 00:07:11,360 --> 00:07:17,030 So we can contrast that with this kind of function, which we might have learned about in an algebra 95 00:07:17,030 --> 00:07:17,570 course. 96 00:07:17,570 --> 00:07:23,600 In this kind of function, we can find negative values of F of X, For instance, if we were to set 97 00:07:23,600 --> 00:07:29,750 X equal to one half, our input would be one half and our output would be one half minus one or negative 98 00:07:29,750 --> 00:07:30,320 one half. 99 00:07:30,320 --> 00:07:32,450 And we see that along the graph here. 100 00:07:32,450 --> 00:07:39,860 If we pick this x equals one half value and we connect it to our graph and then go straight over to 101 00:07:39,860 --> 00:07:44,990 the vertical axis, we see that we get a negative value of one half and F of x. 102 00:07:44,990 --> 00:07:49,220 This function is perfectly defined for that set of values. 103 00:07:49,220 --> 00:07:53,420 We could say the input is one half, the output is negative one half. 104 00:07:53,420 --> 00:07:55,250 That's totally valid. 105 00:07:55,250 --> 00:08:02,600 That satisfies this function f of x, but for probability mass functions we can never find a negative 106 00:08:02,600 --> 00:08:04,040 output like this. 107 00:08:04,040 --> 00:08:08,570 In our case, that means we can never get a negative probability. 108 00:08:08,570 --> 00:08:13,160 So all of these values right here have to be zero or positive. 109 00:08:13,160 --> 00:08:19,790 They cannot be negative, which makes sense because negative probability is really nonsensical. 110 00:08:20,000 --> 00:08:21,980 So those are our three requirements. 111 00:08:21,980 --> 00:08:26,810 We have to be dealing with a discrete random variable, which means we have a countable set of values. 112 00:08:26,810 --> 00:08:28,940 Our set of values is not continuous. 113 00:08:28,940 --> 00:08:30,800 We don't have a continuous random variable. 114 00:08:30,800 --> 00:08:36,980 And all of these probability outputs here have to be defined between zero and one, and in total they 115 00:08:36,980 --> 00:08:38,270 have to add to one. 116 00:08:38,270 --> 00:08:41,330 By definition, of course, that means they can't be negative. 117 00:08:41,360 --> 00:08:47,090 So with that said, the only other thing we want to say about probability mass functions is that sometimes 118 00:08:47,090 --> 00:08:52,310 we'll see them written this way and then sometimes we'll add an extra line to the function, or we'll 119 00:08:52,310 --> 00:08:59,210 see this line added to the function that says that the function is zero for any other values or zero 120 00:08:59,210 --> 00:09:00,020 otherwise. 121 00:09:00,020 --> 00:09:05,720 And if we have this line, all it's doing is clarifying for us that the probability that X takes on 122 00:09:05,720 --> 00:09:09,830 any other value other than zero one or two is zero. 123 00:09:09,860 --> 00:09:14,690 It doesn't really change anything because our probabilities still add to one. 124 00:09:14,690 --> 00:09:20,990 We're just explicitly stating by putting in this extra piece of information that these are indeed the 125 00:09:20,990 --> 00:09:27,200 only values that X can take and confirms for us that the total probability of this probability mass 126 00:09:27,200 --> 00:09:30,200 function is defined in these first three rows here. 127 00:09:30,200 --> 00:09:32,420 So sometimes we'll see this, sometimes we won't. 128 00:09:32,420 --> 00:09:34,160 We don't necessarily have to have it. 129 00:09:34,160 --> 00:09:38,930 It's just important that we meet all of the other requirements of discrete random variable probabilities 130 00:09:38,930 --> 00:09:42,500 defined between zero and one that all sum to one. 131 00:09:42,500 --> 00:09:48,860 Now, just like with this function, this algebraic function where we were able to sketch its graph, 132 00:09:48,860 --> 00:09:52,100 let's talk about the graph of a probability math function. 133 00:09:52,460 --> 00:09:57,710 So really we're setting it up on horizontal and vertical axes, just like we did over here. 134 00:09:57,710 --> 00:10:04,370 And we always include all possible values of x, our discrete random variable along the horizontal axis. 135 00:10:04,370 --> 00:10:08,390 So here are possible values for x or zero one or two. 136 00:10:08,390 --> 00:10:14,210 So we include those along the horizontal axis zero one and two, and then along the vertical axis we'll 137 00:10:14,210 --> 00:10:17,750 mark off increments of our probability P of X. 138 00:10:17,750 --> 00:10:25,880 And then in order to graph this probability mass function, all we do is add in these columns here which 139 00:10:25,880 --> 00:10:33,410 show that the probability that X is zero is 0.25 or 25% or one in four. 140 00:10:33,410 --> 00:10:41,030 So we draw in this bar, this line at x equals zero and extending up to 0.25, whereas the probability 141 00:10:41,030 --> 00:10:43,780 that x equals one is 0.5. 142 00:10:43,790 --> 00:10:50,780 So at X equals one, we sketch in this bar extending up to 0.5 along the vertical axis and then again 143 00:10:50,780 --> 00:10:56,840 here at x equals two, since the probability associated with x equals two is 0.25, we sketch in this 144 00:10:56,840 --> 00:10:59,450 bar all the way up to 0.25. 145 00:10:59,450 --> 00:11:06,380 So seeing the way that these two things match each other, we should be able to from this function sketch 146 00:11:06,380 --> 00:11:09,110 the graph of the probability mass function. 147 00:11:09,110 --> 00:11:14,270 But we should also be able to take this graph and define an algebraic form up here. 148 00:11:14,300 --> 00:11:16,460 The probability mass function F of x. 149 00:11:16,460 --> 00:11:21,110 Obviously, if we were starting from the graph, we could start here along the horizontal axis to see 150 00:11:21,110 --> 00:11:26,150 that x is defined at zero one or two and we could set up this part. 151 00:11:26,820 --> 00:11:29,940 Of our probability mass function. 152 00:11:29,940 --> 00:11:35,760 And then for each of those values, zero one and two, we would find the corresponding probability and 153 00:11:35,760 --> 00:11:37,020 we would set up. 154 00:11:37,710 --> 00:11:42,990 These pieces and then we would just say F of X is equal to we'd have our open brace here. 155 00:11:42,990 --> 00:11:49,620 And then our three scenarios defining the probability for each possible value of the discrete random 156 00:11:49,620 --> 00:11:50,070 variables. 157 00:11:50,070 --> 00:11:56,490 So we should be able to move back and forth between this algebraic representation and this visual graphical 158 00:11:56,490 --> 00:11:57,480 representation. 159 00:11:57,660 --> 00:12:02,610 And then the only other thing we want to say is that just like when we talked about probability before 160 00:12:02,610 --> 00:12:08,340 and we learned about complimentary probability and and or probability, we can answer those same kinds 161 00:12:08,340 --> 00:12:10,590 of questions with a probability math function. 162 00:12:10,590 --> 00:12:17,400 So whether we have this representation of F of X or this graphical representation of F of X, we should 163 00:12:17,400 --> 00:12:22,350 of course be able to state the probability that X is equal to zero or x is equal to one or x is equal 164 00:12:22,350 --> 00:12:23,070 to two. 165 00:12:23,070 --> 00:12:28,290 But we should also be able to find the probability, let's say, that X is equal to zero or one. 166 00:12:28,290 --> 00:12:35,010 We know from the addition rule that the probability that X is equal to zero or one has to be 0.25 plus 167 00:12:35,010 --> 00:12:35,820 0.5. 168 00:12:35,820 --> 00:12:49,830 So the probability that x is zero or one would be equal to 0.25 plus 0.5 or 0.75. 169 00:12:49,890 --> 00:12:54,900 Or we could say something like What's the probability that X is not one? 170 00:12:54,900 --> 00:12:59,790 So the probability that X is not one. 171 00:13:00,030 --> 00:13:03,210 Well, that would be everything but this 0.5 here. 172 00:13:03,210 --> 00:13:06,450 So we would take 0.25 plus 0.25. 173 00:13:07,500 --> 00:13:16,950 And we would get 0.5 or the probability of not one we could calculate as one minus the probability of 174 00:13:16,950 --> 00:13:19,950 x equal to one or 0.5. 175 00:13:19,950 --> 00:13:22,570 And we get the same answer, 0.5. 176 00:13:22,590 --> 00:13:28,830 And of course, we could answer these kinds of questions from this representation of F of X, but also 177 00:13:28,830 --> 00:13:35,040 from the graphical representation of F of X, for instance, this probability of zero or one question. 178 00:13:35,040 --> 00:13:41,460 We would just identify here the probability as 0.25 and here the probability as 0.5. 179 00:13:41,460 --> 00:13:44,280 We would add those together to get 0.75. 180 00:13:44,400 --> 00:13:50,370 So that's the idea of a probability math function, how they're related to the original idea of a function 181 00:13:50,370 --> 00:13:56,910 as a set of inputs and outputs, how the outputs are defined as measures of probability, how we can 182 00:13:56,910 --> 00:14:02,340 only use them for discrete random variables where the probabilities are all defined between zero and 183 00:14:02,340 --> 00:14:08,610 one and sum to one, and how we can represent probability mass functions both algebraically and graphically, 184 00:14:08,610 --> 00:14:13,560 and use both representations to answer all different kinds of probability questions.