Question

The probability density function for a uniform random variable on the interval [a, b] is given by f(x) = { 1/(b-a) a

Answer 1

The question revolves around the properties and calculations related to a uniform distribution, a probability distribution where all outcomes are equally likely within a certain interval. It involves understanding the probability density function, cumulative distribution function, and standard deviation for the continuous random variable.

Understanding the Uniform Distribution

Certainly! To find the cumulative distribution function (CDF) for a uniform random variable on the interval

[a, b], you can follow these steps:

1. Understand the Probability Density Function (PDF):

The given PDF is f(x)=
`(1)/(a-b)` for a≤ x ≤b, and f(x)=0 elsewhere.

2. Write down the CDF definition:

The cumulative distribution function (CDF) is defined as the probability that the random variable takes a value less than or equal to a given

F(x)=P(X ≤ x)

3. Find the CDF for x<a:

For any x<a, the probability is 0, so:

F(x)=0 for x<a

4. Find the CDF for a ≤ x ≤b:

For a ≤x ≤b, integrate the PDF from a to x:

F(x)=
`\int\limits^x_a`
`(1)/(a-b)` dt

F(x)=
`(1)/(a-b)`
`\int\limits^x_a` 1dt

F(x)=
`(1)/(a-b)`
`[t]^x_a`

F(x)=
`(1)/(a-b)` (x−a)

5. Find the CDF for x>b:

For any x>b, the probability is 1, so:

F(x)=1 for x>b

So, the cumulative distribution function

F(x) for a uniform random variable on the interval

[a, b] is given by:

0 & \text{for } x < a \\

\frac{x - a}{b - a} & \text{for } a \leq x \leq b \\

1 & \text{for } x > b

\end{cases} \]

This function describes how the probability accumulates as \( x \) increases within the specified interval.