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In summary: So you compute the probability of x being between 0,9 and 1,1, you get a non zero value but if you take a smaller interval like 0,99 to 1,01 you will get a smaller probability but still non zero. So the probability tends to zero but never reaches zero. The reason is that we are dealing with an uncountably infinite number of points and not with a countably infinite number of points.In summary, the probability of a continuous random variable taking on a particular value is always zero. However, the probability of it falling within a range can be nonzero and is determined by integrating the probability density function over that range. This is because we are dealing with an uncountably infinite set of

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Ray Vickson

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theone said:## Homework Statement

Can someone explain why f(x) = 1/(b-a) for a<x<b ?

## Homework Equations

## The Attempt at a Solution

shouldn't it be 0? since its a continuous random variable and so that interval from a to b has an infinite number of possible values?

What do YOU think ##f(x)## means in this context? (Perhaps you are mis-interpreting the symbols, etc.)

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theone

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Ray Vickson said:What do YOU think ##f(x)## means in this context? (Perhaps you are mis-interpreting the symbols, etc.)

the probability that a continuous random variable X takes on one of its possible values x?

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theone

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Diegor said:The probability to get a particular value for X is 0 but if you want to know the probability to get X within a range, for example 3<X<4 you need to integrate the probability density between these two values and that is not always 0.

to get the probability of x falling within a range, aren't you essentially adding the probabilities of x taking on the particular values within the range... but if the probability of x taking on a particular value is 0, then why is this sum not always zero?

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Ray Vickson

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theone said:the probability that a continuous random variable X takes on one of its possible values x?

No, absolutely not. For a continuous random variable the probability that ##X = x## is zero for all ##x.## However, the probability that ##X## takes a value between ##x## and ##x+\Delta x## (for small but nonzero ##\Delta x##) is given by ##f(x) \Delta x## (to lowest order in ##\Delta x##. For a uniform distribution on the interval ##[a,b]##, we have

$$ P\{ c < X < d \} = \frac{d-c}{b-a} = \frac{\text{small length}}{\text{large length}}$$

for any ##a \leq c < d \leq b##.

In other words, for a uniform distribution over an interval the probability of a sub-interval is proportional to the length of that sub-interval, but is independent of its "location". We get the same probability whether the sub-interval is near the beginning, in the middle, or near the end of the main interval.

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Ray Vickson

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theone said:to get the probability of x falling within a range, aren't you essentially adding the probabilities of x taking on the particular values within the range... but if the probability of x taking on a particular value is 0, then why is this sum not always zero?

theone said:to get the probability of x falling within a range, aren't you essentially adding the probabilities of x taking on the particular values within the range... but if the probability of x taking on a particular value is 0, then why is this sum not always zero?

Because we have an uncountably infinite set of points. The probability of an interval is NOT a SUM of point probabilities!

If I take the interval from 0 to 1 would you say its length must be zero because it is made up of points having length zero? If I take a triangle of base 2 and height 1 would you say its area is zero because it is made up of points of area zero? Well, probability behaves just like length or area in that regard.

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Diegor

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Think this way: Suppose that the probability to find values of x in a range let's say between 3 and 4 is not zero and then you take a small interval like 3,5 to 3,51, you will see that the probability is smaller. If you continue to shrink the interval you will find that the probability tends to zero but still you have a non zero probability in the (3,4) interval.

For example you go to the market to buy one liter of milk. It is unlikely that you find one bottle with exactly one liter but within 0,9 to 1,1 liters surely you will get more than one.

A continuous uniform distribution function is a probability distribution that describes the likelihood of a continuous random variable being within a certain range. It is also known as a rectangular distribution, as the probability density function is a flat line within the specified range.

The characteristics of a continuous uniform distribution function include a constant probability density function within a given range, a total area under the curve of 1, and equal probabilities for all values within the range.

A continuous uniform distribution function is used for continuous random variables, while a discrete uniform distribution is used for discrete random variables. In a continuous uniform distribution, the probability of any specific value is 0, while in a discrete uniform distribution, all values have equal probabilities.

The continuous uniform distribution function is commonly used in statistics and probability to model situations where all outcomes within a range are equally likely. This includes applications in finance, engineering, and physics.

To calculate probabilities using a continuous uniform distribution function, you need to know the range of values for the random variable and the specific value or range of values you are interested in. The probability is then calculated by finding the area under the curve within the specified range.

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