# Probability given a cumlative funcion

• kuahji
In summary, the conversation involves a discussion about determining the distribution function F(x) for a given probability density function and solving for the probability of X being greater than 10. The original function is correctly determined, but there is confusion surrounding the probability of P(X>10). The professor initially states that it should be 0, but later says it should be 1 because there is no upper bound for the given function. However, this does not make sense as it contradicts the solution for P(X<6). The conversation ends with the suggestion to clarify with the professor.

#### kuahji

First determine the distribution function F(x)

f(x)= x/16 for 0<x<4
1/2-x/16 for 4$$\leq$$x<8
0 for elsewhere

So I determined this & came up with the function (which is correct)

F(x)= 0 for x<o
x^2/32 for 0<x<4
x/2-x^2/32-1 for 4$$\leq$$x<8
1 for x$$\geq$$8

Then there is the question P(X>10)

Now, my thoughts is that from the original problem, the probability that x is greater than 8 is zero, & therefore, it should be zero... But, I got the question wrong & the professor stated that P(X>10) is 1. Which doesn't make a whole lot of sense to me, but she explained it that since it doesn't have an upper bound, it must be 1. Any ideas here? We had a homework question that stated a similar P(11<x<12) but here, it was zero because she stated it was bounded outside of the range.

I'm with you. P(X>10) is 0. P(X<10) is 1. I don't see how you could interpret the problem otherwise.

Do you think given that it is a big X & not a little x has anything to do with it? I could see if it was F(X>10) perhaps... the notation always ends up getting the best of me.

Could you state the problem exactly it is given? Yes, P(x> 10) is 0 but you titled this "cumulative" function and have F(x)= 1 for $x\ge 8$.

It is stated "Determine the distribution function, F(x) of the random variable X whose probability density is..." what I outlined first.

It then says "Use the information about F(x) in the previous question to determine" (which is the cumulative function I outlined)
a) P(X<6) which I did & showed the work 6/2-36/32-1 = 7/8 (which was correct)
b) P(X>10) which I showed the work 1-1=0 (which was marked wrong)
in class she specifically said b) was suppose to be 1. 1-0=1

I think this is all a misunderstanding. If P(X<6) is 7/8 then P(X>10) CANNOT be 1. They add to a number larger than 1 and they are mutually exclusive. That would just plain be silly. Please ask your teacher to explain. I can't.

## What is a cumulative function?

A cumulative function is a mathematical concept used to describe the probability of an event occurring within a specified range. It is the sum of the probabilities of all events occurring up to a certain point in a probability distribution.

## How is probability calculated using a cumulative function?

To calculate probability using a cumulative function, you need to have a probability distribution and a specified range. You then use the cumulative function to find the sum of all probabilities within that range.

## What is the difference between a cumulative function and a probability distribution?

A probability distribution is a function that assigns probabilities to different outcomes of an event. A cumulative function, on the other hand, is the sum of probabilities up to a certain point in the probability distribution.

## Why is a cumulative function useful in probability calculations?

A cumulative function allows us to calculate the probability of an event occurring within a specific range, rather than just the probability of a single event. This can be useful in various applications, such as in predicting stock prices or analyzing risk in decision making.

## How do you interpret the results of a cumulative function?

The results of a cumulative function represent the probability of an event occurring within a specified range. For example, if the result is 0.6, it means that there is a 60% chance that the event will occur within that range.