P(a1<X<a2, b1<Y<b2) Understanding

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In summary: Your Name]In summary, the given identity, P(a1<X<a2, b1<Y<b2) = F(a1,b1)+F(a2,b2)-F(a1,b2)-F(a2,b1), can be proved using set theory and probability identities such as P(AUB) = P(A) +P(B) - P(AB). This can be done by defining events A = {a1<X<a2} and B = {b1<Y<b2} and rewriting the identity in terms of these events and the joint cumulative distribution function (cdf). This approach helps in understanding the geometric interpretation of the identity and demonstrates your initiative in exploring and understanding concepts beyond what is required. Keep
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silvermane
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Homework Statement



My professor showed us the identity,
P(a1<X<a2, b1<Y<b2) = F(a1,b1)+F(a2,b2)-F(a1,b2)-F(a2,b1)
where (X,Y) are jointly distributed rvs with a joint cdf of F(x,y) = P(X[tex]\leq[/tex]x, Y[tex]\leq[/tex]y) and a1<a2, b1<b2.
It's not homework that we turn in, but a supplement that she showed us to think about and look at.

The Attempt at a Solution


I understand what it is geometrically with integration, however I want to see how it could be proved using set theory and probability identities such as
P(AUB) = P(A) +P(B) - P(AB), etc.

Thanks so much for your help in advance! :blushing:
 
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Thank you for sharing your professor's identity and your attempt at understanding it. It is always great to see students taking the initiative to understand concepts beyond what is required for homework.

To prove this identity using set theory and probability identities, we can start by defining the events A = {a1<X<a2} and B = {b1<Y<b2}. Then, we can rewrite the given identity as:

P(A∩B) = P(A) + P(B) - P(A∪B)

Using the definition of joint cumulative distribution function (cdf), we can rewrite P(A∩B) as F(a2,b2) - F(a1,b2) - F(a2,b1) + F(a1,b1). Similarly, P(A) and P(B) can be written as F(a2,∞) - F(a1,∞) and F(∞,b2) - F(∞,b1), respectively. Finally, P(A∪B) can be written as F(a2,∞) - F(a1,∞) - F(∞,b1) + F(a1,b1).

Substituting these values into the rewritten identity, we get:

F(a2,b2) - F(a1,b2) - F(a2,b1) + F(a1,b1) = F(a2,∞) - F(a1,∞) + F(∞,b2) - F(∞,b1) - F(a2,∞) + F(a1,∞) + F(∞,b1) - F(a1,b1)

Simplifying, we get:

F(a2,b2) = F(a2,∞) + F(∞,b2) - F(∞,b1) - F(a1,∞) + F(a1,b1)

which is the same as the given identity.

I hope this helps in understanding the proof of this identity. Keep up the good work in exploring and understanding concepts beyond what is required. That is what makes a great scientist.

 

FAQ: P(a1<X<a2, b1<Y<b2) Understanding

1. What does "P(a1<X<a2, b1<Y<b2)" mean?

The notation "P(a1<X<a2, b1<Y<b2)" represents the probability of a random event occurring within a specific range for two variables, X and Y. The first part of the expression, "a1<X<a2", means that the value of X falls between a1 and a2. Similarly, "b1<Y<b2" indicates that Y falls between b1 and b2. Therefore, the entire expression represents the probability of both X and Y falling within their respective ranges.

2. How is "P(a1<X<a2, b1<Y<b2)" different from "P(X=a, Y=b)"?

The first expression, "P(a1<X<a2, b1<Y<b2)", represents the probability of a range of values for both X and Y, while the second expression, "P(X=a, Y=b)", represents the probability of specific values for X and Y. In other words, the first expression allows for a range of values, while the second expression is more specific.

3. What is the relationship between "P(a1<X<a2, b1<Y<b2)" and "P(X<a, Y<b)"?

Both expressions involve probabilities for two variables, X and Y. However, "P(a1<X<a2, b1<Y<b2)" represents the probability of both X and Y falling within a specific range, while "P(X<a, Y<b)" represents the probability of both X and Y falling below a certain value. In other words, the first expression is more restrictive, as it requires both X and Y to fall within their respective ranges, while the second expression allows for more flexibility in the values of X and Y.

4. How can "P(a1<X<a2, b1<Y<b2)" be calculated?

The calculation of "P(a1<X<a2, b1<Y<b2)" involves finding the area under the probability density function (PDF) curve for both X and Y within their respective ranges. This can be done using various methods, such as integration or numerical approximation techniques.

5. What is the significance of "P(a1<X<a2, b1<Y<b2)" in scientific research?

The expression "P(a1<X<a2, b1<Y<b2)" is commonly used in scientific research to represent the probability of a specific outcome or event occurring within a specific range for two variables. It allows researchers to quantify the likelihood of certain results and draw conclusions based on statistical significance. This expression is often used in fields such as biology, psychology, and economics to analyze data and make informed decisions.

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