Cumulative distribution function question

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).
  • #1
sneaky666
66
0
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?
 
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  • #2
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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