Proof of a formula with two geometric random variables

In summary, the formula for calculating the probability of two geometric random variables is P(X=x, Y=y) = P(X=x) * P(Y=y). This formula can be interpreted as the probability of X taking on a specific value x AND Y taking on a specific value y at the same time. If the two random variables are independent, the formula can be simplified to P(X=x)*P(Y=y). To find the probabilities of two geometric random variables, you need to know the individual probabilities of each variable and then multiply them together using the formula P(X=x, Y=y) = P(X=x) * P(Y=y). However, this formula is only applicable to two geometric random variables and may not be used for other types of
  • #1
Armine
2
0
Homework Statement
If G_1 and G_2 are independent geometric random variables with parameters p_1 and p_2 respectively, show that
Relevant Equations
P(G_1<G_2)=p_1(1-p_2)/(p_1+p_2-p_1p_2)
The image above is the problem and the image below is the solution I have tried but failed.

MVIMG_20210209_102231_recompress.jpg
1612521305080_recompress.jpg
 
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  • #2
You first equation is multiple-counting many cases. You should sum the cases where ##G_1 = k##, not ##G_1 \le k##
 
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  • #3
FactChecker said:
You first equation is multiple-counting many cases. You should sum the cases where ##G_1 = k##, not ##G_1 \le k##
Get it, thank you very much!
 

Related to Proof of a formula with two geometric random variables

1. What is a geometric random variable?

A geometric random variable is a discrete random variable that represents the number of trials needed to achieve a success in a sequence of independent trials, where each trial has a constant probability of success.

2. What is a proof of a formula with two geometric random variables?

A proof of a formula with two geometric random variables is a mathematical demonstration of the relationship between two geometric random variables and their associated probabilities, using the properties and principles of probability theory.

3. How is the formula for two geometric random variables derived?

The formula for two geometric random variables is derived by applying the laws of probability to the joint distribution of the two variables, and using the definition of a geometric random variable and the properties of independent events.

4. Can the formula for two geometric random variables be applied to other types of random variables?

No, the formula for two geometric random variables is specific to geometric random variables and cannot be applied to other types of random variables, as it is based on the specific properties and probability distributions of geometric random variables.

5. What are some real-world applications of the formula for two geometric random variables?

The formula for two geometric random variables can be applied in various fields, such as finance, biology, and engineering, to model and analyze sequential processes with a constant probability of success, such as stock market fluctuations, genetic mutations, and equipment failures.

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