Find the density function for a random variable that measures the time of a video.

Question

I hope can give me a hint for this exercise.

On a new digital platform such as youtube, each user posts videos of up to 30 seconds. The platform created a remuneration system (by video) that is based on the total viewing time in the first week of publication (of users other than the author of the video). If, for example, a 30-second video is viewed exactly 10 times in full during the first week of publication, the total time would be 300 seconds. If, on the other hand, such a video is viewed only once and halfway through the first week, the total viewing time would be 15 seconds. Let's say that a new video is published and let T be the random variable which represents its total viewing time (in the first week of publication).

(i) Does it make sense to assume that T is a continuous random variable?Justify your answer

Ans: According to the problem, T is the total time, since there are an infinite number of possible times that can be taken it is continous.

(ii) Assuming that T is continuous, choose a set of values ​​that the random variable T should assume??

Ans: That depends on the times of views, so it can take $0\leq T<\infty$

(iii) Propose a density function for T that is compatible with the answer above.

This confusses me because T depends in the time of view, so it will be something like $T(X)$, but it shouldn't be sometinhg like $f(T)$?

(iv) Given the density function that fits with the item (iii), what is the probability that the total video viewing time (in the first week) is less than 60 seconds?

If I understand item (iii) I think can do item(iv)

Thanks

Answer 1

The question is open ended, but here are some initial thoughts:

1. We say $T$ is a continuous random variable when its image is uncountably infinite. Since $T$ represents time, it seems reasonable to assume it is continuous. Note that this does not necessarily imply $T$ is absolutely continuous, which means it has a density (and hence assigns mass zero to any particular time). Not sure if you have learnt of this distinction as yet, but it is a technicality.

2. A lower bound is zero, but it is not clear that the upper bound should be infinite. There are only 604,800 seconds in a week---multiply that by the number of possible distinct viewers. Nonetheless, we can choose $T\in [0,\infty)$ for modeling simplicity.

3. Assuming $T$ is absolutely continuous, choose a density that best represents the data. There are nonparametric ways to do this (e.g. kernel density estimation), but it looks like you want a parametric approach here. Any density with a support of $[0,\infty)$, e.g. a chi-squared distribution, is then technically feasible so long as you can justify it against the data. The density takes the form $f(t)$ where $f(t)\geq 0\quad\forall t\in [0,\infty)$ and $\int_0^\infty f(t)=1.$

4. $P(T<60)=\int_0^{60} f(t)dt$.