00:01
In this problem, we need to calculate the probability of no logons and also we need to find out the median between median time between logons.
00:08
Let us see how we can do that.
00:09
First of all, we can write there are, there are 25 logons per 60 minutes.
00:17
So it can be written logons per 60 minute.
00:21
Now, we need to see that what will be the number of logons per minute.
00:28
So number of logons per minute will be number.
00:32
We can write this one as number of logons equals 25 over 60 and this will be per minute.
00:46
That's very simple.
00:47
That's the unitary method, isn't it? now we can write that let x be the random variable denoting number of logons on logons in 60 in six minutes.
00:58
So x is our random.
01:01
Variable denoting logons in six minutes random variable so we can write the six minute value is as we can show this one as prox belongs to poison distribution here lambda value that is parameter value is 25 over 60 multiplied six so it will be six and then here we have okay so we can show multiplication as such and then we have here upon simplification p and lambda value will be 2 .5 so from here what we can do we can say that so in six minutes if there is no log on then we are interested to find probability of x equals zero because if there are no logons then we are interested to find this value only and if the formula is probability of capital x equals small x equals e -d -to -the -power minus lambda, lambda raised to the power x and in the denominator we will have x factorial.
01:59
So with this formula, if we plug in the values here, we will have e raised to the power minus 2 .5, lambda that is 2 .5, raised to the power 0 because the value of x is 0 and in the denominator we will have 0 factorial .0 factorial has the value 1.
02:13
So this expression when simplified will give us 0 .08 to 1 as our answer.
02:20
So this is the answer for the first part of the problem.
02:22
Now let us look to the answer or to the process of second part.
02:27
So in the second part, we need to calculate the median.
02:31
So we can firstly say that probability of no log on during time interval t.
02:35
So for that, we need to first write the t lambda value, which was 25 over 60, multiplied t.
02:42
We have already seen this value in the above part.
02:45
Now here we have the formula, isn't it? so based on this particular formula, we can write over here that it will be probability of x equals small x and this will be e raised to the power minus lambda lambda raised to the power x over x factorial so here we need to understand one thing that the value of lambda will be 25 over 60 that when solved will give us 0 .4167 t so don't get confused this value is point 4167 now lambda value can also be written over here 0 .4167 raise to the power x okay we missed t so in the denominator we have x factorial now we need to calculate the probability of no log on during time interval t so it will be probability of x equals zero so our final equation will look like e -r -dose -to -the -power negative point 4167 because zero factorial is 1 and any any number or any base having the power as 0 will be equals to 1 so we have been we have only e -dise to the power negative 0 .4167 multiplied t now we know that for me what is the value given to us? we know that for median the probability of x equals 0 is 0 .5...