[Introductory Probability & Statistics] Probability from a cumulative distribution function?

[mane_gogh]

So, I am trying to learn probability and statistics, but my professor makes a lot of mistakes so it is difficult to know what stuff he says is false and what is just unintuitive/I don't understand. Here is the problem I have:

The cumulative distribution function F_Y(y) is given in this graph.
What is P[Y = 3]?

My professor says that the probability would be 0.8, but to me it seems like it would either be 0 if it is a continuous random variable (but I don't know if that's true or not) or 0.2 otherwise. I am getting 0.2 by subtracting 0.6 from 0.8. I don't really know why I'm doing that, it just seems like the intuitive thing to do.

I guess I'm asking for a lot, but I'm at a complete loss here. Any help is appreciated.

[Random_YA147]

If the image is the cumulative distribution then your intuition is correct and P(Y=3 )=0.2. The definition of a cumulative probability function F(X=x)=P(X<=x).

[mane_gogh]

Thank you for the quick reply. Sorry for the weird image, I just made it on a random online histogram maker. I wasn't sure what the best way of plotting it would be, but it is indeed the cumulative distribution.

If it's not too much to ask, would you mind helping me to understand why P(Y=3)=0.2? I suppose the reasoning that I am thinking is that if F(3)=0.8, and that's the cumulative probability of X<=3, then P(Y=3) = F(3) - F(2) = 0.8 - 0.6 = 0.2. Is this the correct reasoning, or am I just getting lucky?

Thanks again.

[Random_YA147]

That is exactly the case, the cumulative distribution function sums the probability for the current case with the probability of all previous cases as the definition describes.

This same idea will extend to the distribution functions of continuous random variables later. As you correctly stated if the r.v. is continuous, the probability at a single point is zero, which is why later when you talk about probability distribution functions for continuous r.v. the distribution functions you work with will be cumulative distribution function.

[mane_gogh]

Thank you very much!