# Uniform Distribution: PDF, Mean, and Standard Deviation

• rclakmal
In summary, a continuous random variable X is uniformly distributed in the interval 2 <= X <= 6 and can take no other value. The probability density function is f(x) = 1/4. The mean is 4 and the standard deviation is 4/3. The distribution function, or cumulative density function (CDF), is F(x) = x/4 - 1/2. The CDF measures the probability up to a certain point of X, instead of just at that point.

#### rclakmal

Unifrom distribution ?

a continuous random variable X is uniformly distributed in the interval 2<=X<=6 and can take no other value
1.what is the probability density function ?
2.find mean and standard deviation of X ?
3.obtain distribution function of X?

yr obviously i can answer for first two question ...

1.for a uniform function
PDF is f(x)=1/(b-a)====>f(x)=1/4

2.mean=a+b/2=2+6/2=4;
SD= (b-a)^2/12=4^2/12=16/12=4/3

but i have no idea how to do the third part ...please can anybody help me out...what is distribution function of X ...Not only the answer ...theory behind it ...then i may be able to get the answer for my self

Look up the uniform distribution function

i don't understand u mate ...can u explain it a bit !it will be really helpful thanks!

hey no help for 3 hours ...i have provided all ma works and waiting for someone to help!please tell me what is the distribution function ...how it is ddifer from PDF ?any idea ?//and how to approach it ?

Which of the words in "Look up uniform distribution function" did you not understand?
You seem to not know what a "distribution function" is. Surely that definition is given in your textbook or, if not, on the internet.

If you had done that you would have found that the "probability density function" is the derivative of distribution function and that the distribution function, the probability that the result is between a and b, is the integral, from a to b, of the density function.

yr thanks !i did think like that ...but i was getting 1 as the answer ...is it correct?if it is correct then all the uniform distributions have the same distribution function that is 1...if that so thanks for helping me ...

You misinterpreted what I said. In fact, now that I look at it, I didn't say it very well! The "distribution function", F(x), gives the probability a value of between a and b by F(b)- F(a). F(x) itself is the anti-derivative of f(x) the constant of integration determined by F(2)= 0. In other words, it is $F(x)= \int_2^x f(x) dx$.

Ok then answer should be F(x)=x/4-1/2; am i correct now ...ii think so ...so i have another question ...is it the same function which is caled as cumulative density function (CDF)??

rclakmal said:
Ok then answer should be F(x)=x/4-1/2; am i correct now ...ii think so ...so i have another question ...is it the same function which is caled as cumulative density function (CDF)??

The F(X) is called the CDF or cumulative distribution function. What this means is that the CDF measures P(X <= A) instead of the usual PDF measure which is P(X = A) or as in the continuous case P(A < X < B).

So the CDF measures probability up till a certain point of X. So its like saying that if we wanted to find P(X < A) we sum all probabilities up to but less than A instead of say at just A.

## 1. What is a uniform distribution?

A uniform distribution, also known as a rectangular distribution, is a probability distribution where all possible outcomes have an equal chance of occurring. This means that the data is evenly spread out over a range, with no particular data point being more likely than another.

## 2. How is a uniform distribution different from other types of distributions?

In contrast to a uniform distribution, other distributions, such as normal or exponential distributions, have specific shapes and patterns that describe the data. A uniform distribution does not have any particular shape or pattern and is considered to be a special case in statistics.

## 3. What is the importance of the uniform distribution in statistics?

The uniform distribution is important in statistics because it provides a baseline or reference point for comparing other distributions. It is also commonly used in simulations and modeling as it represents a situation where all outcomes are equally likely, which can be useful in certain scenarios.

## 4. How is a uniform distribution represented graphically?

A uniform distribution is typically represented graphically using a histogram or a bar graph. The x-axis represents the range of possible outcomes, while the y-axis shows the frequency or probability of each outcome occurring. In a uniform distribution, the bars or columns are of equal height, indicating the equal likelihood of each outcome.

## 5. Can a uniform distribution have different shapes or patterns?

No, a uniform distribution is defined by its equal probability of all outcomes and does not have any particular shape or pattern. However, it can be transformed using mathematical operations to fit a specific range of data or to create a different distribution, such as a normal distribution.