# Cumulative distribution function question

• sneaky666
In summary, we are considering rolling a die and defining the random variables X and Y. To compute the cumulative distribution function Fy(y) for Y, we can use the formula P(Y<=y) = Fx(y) - Fx(-inf). For example, for Y=1, we get Fy(1) = Fx(1) - Fx(-inf) = Fx(1). We can also use the standard normal distribution to compute probabilities. For example, to find P(X<=-5), we can use the standard normal table or a calculator to evaluate ϕ(-5).
sneaky666
consider rolling a die.
S= {1,2,3,4,5,6}
P(s)=1/6 for all s in S
X= number on die so that X(s)=s for all s in S
Y= X^2
compute the cumulative distribution function Fy(y) = P(Y<=y), for all y in the set of real numbers.

My guess
for Y=1 i get
P(-inf<y<=1)=P(Y<=1)-P(Y<-inf)=Fx(1)-Fx(-inf)
= Fx(1)-0
= Fx(1)

Is this all I have to do for Y=1, or do I have to integrate, or is there anything wrong?

EDIT: ok i figured it out but i need help on this one.

Let X~N(0,1) . Compute each in terms of function ϕ.
And evaluate it numerically.

P(X<=-5)
P(-2<=X<=7)
P(X>=3)

for the first one i get
ϕ(-5)

But how do i evaluate it?

## 1. What is a Cumulative Distribution Function (CDF)?

A Cumulative Distribution Function (CDF) is a statistical function that maps the probability of a random variable X being less than or equal to a specific value. It is a convenient way to describe the distribution of a dataset and is commonly used in probability theory and statistics.

## 2. How is a CDF different from a Probability Distribution Function (PDF)?

A CDF is the cumulative sum of probabilities for all values of the random variable X, while a PDF is the probability of a single value of X occurring. In other words, a CDF gives the probability of X being less than or equal to a certain value, whereas a PDF gives the probability of X taking on a specific value.

## 3. What is the purpose of using a CDF in statistical analysis?

A CDF can provide valuable insights into the distribution of a dataset, such as the likelihood of certain values occurring and the shape of the distribution. It can also be used to calculate various statistical measures, such as percentiles and quartiles.

## 4. Can you explain the relationship between a CDF and a Probability Density Function (PDF)?

The CDF is the integral of the PDF, meaning that the area under the PDF curve up to a specific value is equal to the value of the CDF at that point. In other words, the CDF is the cumulative sum of the probabilities represented by the PDF.

## 5. How is a CDF used in hypothesis testing?

In hypothesis testing, the CDF is used to determine the probability of obtaining a certain sample mean or proportion, assuming the null hypothesis is true. This probability is then compared to a significance level to determine if the null hypothesis should be rejected or not.

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