# Help with cumulative distributions

• sneaky666
The cdf should be an increasing function, not a piecewise function like this. Also, the range of the function should be from 0 to 1, not from -∞ to +∞. Without more context, I cannot provide a summary.
sneaky666

## Homework Statement

suppose Fy(y)=y^3 for 0<=y<1/2 and Fy(y)=1-y^3 for 1/2<=y<=1. Compute these.

1.
P(1/3<Y<3/4)
2.
P(Y=1/3)
3.
P(Y=1/2)

## The Attempt at a Solution

Is this right for the 1. ?
P(1/3<Y<3/4)
P(1/3<Y<1/2) + P(1/2<=Y<3/4)
( Fy(1/2-) - Fy(1/3) ) + ( Fy(3/4-) - Fy(1/2-) )
Fy(1/2-) - Fy(1/3) + Fy(3/4-) - Fy(1/2-)

Now I have to use
Fy(y)=y^3 for 0<=y<1/2 and Fy(y)=1-y^3 for 1/2<=y<=1

1/8 - 1/27 + 37/64 - 1/8
=935/1728

for 2 i am getting 1/27
for 3 i am getting 7/8

I think there's something wrong, but I don't know what.

Is Fy supposed to be the cumulative distribution function? If it is, you must have written down the wrong function because the cdf should be an increasing function. If it's supposed to be a probability density function, there's also a problem because the area under the curve from -∞ to +∞ isn't equal to 1.

the exact question was:

Suppose Fy(y)=y^3 for 0<=y<1/2, and Fy(y)=1-y^3 for 1/2<=y<=1. Compute each of the following:
a) P(1/3<Y<3/4)
b) P(Y=1/3)
c) P(Y=1/2)

the section name is cumulative distribution function
chapter name is random variables and distribution

## 1. What is a cumulative distribution?

A cumulative distribution is a statistical concept that shows the probability of a random variable being less than or equal to a certain value. It is calculated by the sum of all probabilities up to that value.

## 2. How is a cumulative distribution different from a probability distribution?

A probability distribution shows the probability of each possible outcome of a random variable, while a cumulative distribution shows the probability of the random variable being less than or equal to a certain value. In other words, a cumulative distribution is the cumulative sum of probabilities from a probability distribution.

## 3. Why is a cumulative distribution important in statistics?

Cumulative distributions are important in statistics because they provide a visual representation of the likelihood of a certain outcome or range of outcomes. They can also be used to calculate percentiles and determine the probability of events occurring within a specific range.

## 4. How do you interpret a cumulative distribution graph?

A cumulative distribution graph shows the cumulative probability of a random variable on the y-axis and the values of the random variable on the x-axis. The graph can be interpreted by looking at the curve, which represents the probability distribution, and the point on the x-axis, which represents the value at which the cumulative probability is being calculated.

## 5. How can you use cumulative distributions in real-world applications?

Cumulative distributions are commonly used in real-world applications to determine the likelihood of certain events occurring, such as the success rate of a medical treatment or the probability of a stock reaching a certain price. They can also be used to compare different data sets and make predictions based on past trends.

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