00:01
In this question, we are given a cumulative distribution function, and we are asked to calculate the probability mass function from the cumulative distribution function, and also to calculate some probabilities for the random variable x from the cumulative distribution function.
00:19
So in part a, we are asked for the probability mass function of x.
00:28
And recall that for a random variable that can only take on integer values, and that is the case for this question.
00:35
X can only take on integer values.
00:37
It's the number of months.
00:39
These are discrete numbers.
00:42
Then the probability that the random variable takes on x is equal to the cumulative probability at x minus the cumulative probability at x minus 1.
01:02
And so to provide the probability mass function for x, we just have to perform this calculation for all values of x.
01:10
The probability that x takes on the value 1 is the cumulative probability at 1 minus the cumulative probability at 0.
01:30
And from our cdf, the cumulative probability at 1 is 0 .3, that includes the value 1.
01:39
And the cumulative probability at 0, which is x is less than 1, is 0.
01:46
So we have the 0 .3 minus 0 .3 gives us 0 .3 p of 2, it's equal to f of 2 minus f of 1.
02:11
The cumulative probability at x equals 2 is 0 .3, and at 1 is also 0 .3 because both 2 and 1 fall in this range, which pertains to this cumulative probability.
02:33
So we get 0.
02:34
And then we just continue in this manner for all possible values of x.
03:47
Now for p7, p8, p9, p10, and p11, they all are equal to zero.
04:01
Because these values and one minus these values all fall in this range...