Cumalative distribution

I don't get this fam.

I don't even understand the notation.
P(X less than or equal to x).. What's this supposed to mean.

Hoe do you interchange between a cumalitive distribution and a normal P(X=x) distribution?

Man's so lost. :/

Any help is greatly appreciated

Posted from TSR Mobile

I don't get this fam.

I don't even understand the notation.
P(X less than or equal to x).. What's this supposed to mean.

Hoe do you interchange between a cumalitive distribution and a normal P(X=x) distribution?

Man's so lost. :/

Any help is greatly appreciated

Posted from TSR Mobile

There are two ways to represent the probability distribution. One is by means of its mass (or density, in the continuous case) function. That's P(X=x) in the discrete case. The other, which is more easy to represent in the continuous case, is P(X<=x). The reason this is more easy in the continuous case is because P(X<=x) = $\int_0^x f(u) du$ where f is the probability density function. However, the idea of P(X=x) is useless for a continuous distribution, because it's usually 0. Hence the only useful way to calculate probabilities in a continuous distribution is to take the probability that X lies in some interval - and that's equivalent to finding P(X<=x).

That is, $\int_0^x f(u) du$ is the cumulative distribution function; $f(x)$ is the probability density function. Similarly in the discrete case, $\sum_{i=1}^x f(i)$ is the cumulative distribution function, while $f(x)$ is the probability mass function.

I'll use F(x) to signify the probability of X being less than or equal to x and P(x) to signify the probability of X being equal to x.

Let's consider the basic situation where we've got a discrete distribution with two possible values, 1 and 2. If we're given that F(1)=0.5 and F(2)=1, that means P(1)=0.5 and P(1 or 2)=1. Therefore, P(1) = F(1) = 0.5 and P(2) = F(2)-F(1) = 1-0.5 = 0.5.

So in general, to work out P(x), you do F(x)-F(x-1). This works because F(x) is the total probability of everything up until and including x, and F(x-1) is the total probability of everything up until but not including x, so therefore F(x)-F(x-1) is just the chance of x. Bear in mind that when I say F(x-1), I mean F() of whatever the value below x is. For instance, if we've got a distribution of the values 1,3,5, P(5) = F(5)-F(3).

Does that make sense?

Yeah, that's all good my friend. What if you had x = 1,3,5, why is F(3.6) the same as F(3)?

I also don't really understand the notation: P(X=x). I know that it is the probability that your random variable is = to your observed value, but what exactly is a random variable.

Man, S1 is so dry.

Posted from TSR Mobile

what exactly is a random variable.

F(3.6) is not necessarily the same thing as F(3), but it will be in most questions in S1. This is because we're considering discrete distributions where there can only be particular value and generally in S1, these values will be integers just for the sake of simplicity. For instance, if our values are [1,2,3,4,5], then F(3) = P(1,2,3) but F(3.6) is the same thing, since the values that are less than or equal to 3.6 are the same as the values that are less than equal to 3, since there's nothing in between 3 and 3.6.

In a discrete distribution, we've got a set of values, each of which has a set chance of coming up. The event of a value coming up is called P(X), so the event of a particular value x coming up is called P(X=x), i.e. the chance that the value coming up (X) is equal to the particular value x. X is just the symbol given to a value that could come up, and x is just the symbol given to the particular value we're interested in.

Aaah, i see now. Safe. Man was just getting confused with all the Xs and xs. I guess i understand most of this now. Really hope cumalitive doesnt come this year aha, it's appeared in the last two papers.

Posted from TSR Mobile

This is a kind-of-deep question - intuitively, I think of it as a number whose value we don't know yet. We can ask the universe, "what is the value of X at the moment?" and it might answer, for instance "5" - and that's the event X=5. (Much as we could ask the universe, "What is the value of the gravitational acceleration due to gravity at the moment?" and it would answer "9.81 m/s^2".)

The universe runs on strict laws, and they're strict enough that we can work out some things about X's value even if we don't know it exactly. For instance, we might work out that if we asked the universe a hundred times what X's value was, we'd get the answer 5 around 90 times. (That means P(X=5) = 0.9.)

Is that helpful? The actual definition of "random variable" is kind of complicated and not very useful, much like the definition of "rational number".

Yeah, it's not brilliantly explained. If I were you, I would hope for it to come up. In comparison to other things like the normal distribution, it's much more simple in my view. But good luck anyway!

This is a kind-of-deep question - intuitively, I think of it as a number whose value we don't know yet. We can ask the universe, "what is the value of X at the moment?" and it might answer, for instance "5" - and that's the event X=5. (Much as we could ask the universe, "What is the value of the gravitational acceleration due to gravity at the moment?" and it would answer "9.81 m/s^2".)

The universe runs on strict laws, and they're strict enough that we can work out some things about X's value even if we don't know it exactly. For instance, we might work out that if we asked the universe a hundred times what X's value was, we'd get the answer 5 around 90 times. (That means P(X=5) = 0.9.)

Is that helpful? The actual definition of "random variable" is kind of complicated and not very useful, much like the definition of "rational number".