Expected number of random variables that must be observed

In summary, the conversation is about a mathematical equation and a potential typo in the answer given. The expert summarizes that the answer to (a) is correct, while the answer given for (b) is wrong due to a typographical error. The conversation ends with the expert thanking the other person for pointing out the error.
  • #1
WMDhamnekar
MHB
379
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TL;DR Summary
Expected number of random variables that must be observed before any specific sequence.
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In my opinion, answer to (a) is ## \mathbb{E} [N] = p^{-4}q^{-3} + p^{-2}q^{-1} + 2p^{-1} ##
In answer to (b), XN is wrong. It should be XN=p-4q-3 - p-3 q-2- p-2 q-1 - p-1. This might be a typographical error.
Is my answer to (a) correct?
 
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  • #2
WMDhamnekar said:
In my opinion, answer to (a) is ## \mathbb{E} [N] = p^{-4}q^{-3} + p^{-2}q^{-1} + 2p^{-1} ##
Please explain your reasoning.

For b) I agree with you.
 
  • #3
haruspex said:
Please explain your reasoning.

For b) I agree with you.
Answer to (a) given by author is correct. My answer is wrong. Thanks for bringing my error to my notice.
 
  • #4
WMDhamnekar said:
Answer to (a) given by author is correct. My answer is wrong. Thanks for bringing my error to my notice.
You are welcome.
I had never seen this method before. It's brilliant- thanks for posting.
 
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FAQ: Expected number of random variables that must be observed

What is the expected number of random variables that must be observed to achieve a certain outcome?

The expected number of random variables that must be observed to achieve a certain outcome is the average number of observations needed to see that outcome. This is often calculated using the concept of expected value in probability theory, which takes into account the probabilities of different outcomes and their respective counts.

How do you calculate the expected number of observations for a random variable?

To calculate the expected number of observations, you typically use the formula for the expected value of a discrete random variable. For example, if you are looking to observe a specific event with probability \( p \), the expected number of trials needed is given by the inverse of the probability, \( \frac{1}{p} \). This is derived from the geometric distribution.

What role does the geometric distribution play in determining the expected number of observations?

The geometric distribution is crucial in determining the expected number of observations because it models the number of trials needed to get the first success in a series of independent Bernoulli trials (each with success probability \( p \)). The expected value of a geometrically distributed random variable is \( \frac{1}{p} \), which represents the average number of trials needed to achieve the first success.

Can the expected number of observations be applied to continuous random variables?

Yes, the concept of the expected number of observations can be extended to continuous random variables, but the approach may differ. In such cases, you often deal with the expected value of a continuous random variable, which involves integrating the probability density function over the range of possible values. The specific methods may vary depending on the nature of the problem.

What are some practical applications of calculating the expected number of random variables that must be observed?

Calculating the expected number of random variables that must be observed has practical applications in various fields, including quality control (e.g., determining the number of samples needed to detect a defect), finance (e.g., estimating the number of trades needed to achieve a target profit), and biology (e.g., estimating the number of trials needed to observe a genetic trait). Understanding this concept helps in planning and optimizing processes in these areas.

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