Uniform Distribution: What If b-a < 1?

In summary, the function f(x) is equal to 1 divided by the difference between the upper and lower limits of the uniform distribution. However, even if this difference is less than 1, the probability distribution can still take on values greater than one. This does not mean that the probability of an event happening is greater than one, as the actual probability is determined by the integral of the probability distribution over a given range. As long as this integral is not greater than one, the actual probability will never be greater than one.
  • #1
circa415
20
0
If X ~ U(a, b) then f(x) = 1/(b-a)

but what if b-a is less than 1

for instance if X ~ (.5,1) then f(x) = 2?

I'm a bit confused. Any help would be appreciated.
 
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  • #2
A probability distribution p(x) may take on values greater than one. It doesn't mean something has a probability of more than one of happening because, remember, p(x) isn't itself a probability. The probability that X lies in the range x->x+dx is given by p(x)dx, and as long as the integral of this over any range is not greater than one, which is assured by normalization and the fact that p(x) must be non-negative, the actual probability will never be greater than one.
 
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  • #3


If b-a is less than 1, then the interval (b-a) is smaller, meaning that there is less room for the values of X to be distributed. In this case, the probability density function would still be 1/(b-a), but the values of X would be more concentrated and closer together. This means that the probability of getting a specific value of X would be higher compared to when b-a is greater than 1.

For example, if X ~ (0.5, 1), the probability of getting X = 0.6 would be higher than if X ~ (0, 1) because the interval (0.5, 1) is smaller, and therefore, the values of X are more concentrated.

In general, a smaller interval (b-a) would result in a higher probability for each individual value of X, but the total probability of all possible values of X would still be equal to 1. So, in this case, the probability density function would still be valid, but the distribution would be more concentrated around the mean.

I hope this helps clarify any confusion. If you have any further questions, please let me know.
 

What is a uniform distribution?

A uniform distribution is a probability distribution where all possible outcomes have an equal chance of occurring. This means that the probability of any event happening within a certain range is the same.

How is a uniform distribution represented?

A uniform distribution is typically represented graphically as a straight horizontal line, with all possible outcomes having the same height on the graph.

What does it mean if b-a < 1 in a uniform distribution?

If b-a < 1 in a uniform distribution, it means that the range of possible outcomes is less than 1 unit apart. This indicates a narrow and tightly clustered distribution, where the probability of each outcome is very similar.

What are some real-world examples of a uniform distribution?

Some real-world examples of a uniform distribution include rolling a fair die, where each number has an equal probability of being rolled, or selecting a card from a well-shuffled deck, where all cards have an equal chance of being chosen.

How is a uniform distribution different from other distributions?

A uniform distribution is different from other distributions, such as a normal distribution, because it has a constant probability for all possible outcomes, whereas other distributions have different probabilities for different outcomes. Additionally, a uniform distribution is typically used for discrete random variables, while other distributions can be used for continuous random variables.

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