# PROBABILITY: Distribution function

• tamintl
In summary, a distribution function is a mathematical function that shows the probability of a random variable being less than or equal to a certain value. It is different from a probability density function, which describes the relative likelihood of different values of a continuous random variable. Examples of distribution functions include the normal, exponential, and uniform distributions, which are used to describe real-life phenomena. The distribution function is used in probability calculations to find the probability of certain events occurring. It can be used for both continuous and discrete random variables, but for discrete variables, it is a step function instead of a continuous curve.

## Homework Statement

A continuous random variable X has density f(x) = ax2(1- x) for 0 < x < 1, and
f(x) = 0 otherwise. Here a is a constant, to be determined. Find the distribution func-
tion FX, the constant a, the expectation E(X), the variance Var(X), and the conditional
probability p(X < 1/2 | X > 1/4)

## The Attempt at a Solution

Really don't know where to start

Start by using the fact that
$$\int_{-\infty}^\infty f(x)\,dx=1$$
to figure out a. Then use the standard formulas.

## 1. What is a distribution function?

A distribution function, also known as a cumulative distribution function (CDF), is a mathematical function that shows the probability of a random variable being less than or equal to a certain value. It is a way to describe the probability distribution of a random variable and is often used in statistics and probability theory.

## 2. How is a distribution function different from a probability density function?

A probability density function (PDF) is a function that describes the relative likelihood of different values of a continuous random variable. It is the derivative of the distribution function. While the PDF gives the probability of a specific value occurring, the distribution function gives the probability of a value being less than or equal to a certain value.

## 3. What are some examples of distribution functions?

Some examples of distribution functions include the normal distribution function, the exponential distribution function, and the uniform distribution function. These functions are commonly used to describe the probability distribution of real-life phenomena such as height, time between events, and ages.

## 4. How is the distribution function used in probability calculations?

The distribution function is used to calculate the probabilities of certain events occurring. For example, if we want to find the probability of a random variable being between two values, we can use the distribution function to find the probabilities of each individual value and then subtract them to get the desired probability.

## 5. Can the distribution function be used for discrete random variables?

Yes, the distribution function can be used for both continuous and discrete random variables. However, for discrete random variables, the distribution function is a step function that jumps at each possible value of the variable instead of being a continuous curve.