MHB How Do Probability Formulas for Bayes Theorem and Exponential Distribution Work?

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The discussion focuses on understanding the application of probability formulas related to Bayes' Theorem and the exponential distribution. The first formula calculates the probability of a specific event occurring within a given interval, while the second formula illustrates how to apply Bayes' Theorem to find conditional probabilities. Participants express confusion over the calculations and seek clarification on the steps involved. A user acknowledges a previously overlooked detail that simplifies the understanding of the problem. The conversation highlights the importance of breaking down complex probability concepts for better comprehension.
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Hey guys,

I don't understand how this question works... I don't understand the answers either. Could someone take me through this step-by-step?

See attached image:
 

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Longines said:
Hey guys,

I don't understand how this question works... I don't understand the answers either. Could someone take me through this step-by-step?

See attached image:

a) is...

$\displaystyle P \{ G = k\} = \int_{k-1}^{k} e^{- \lambda\ x}\ d x = e^{\lambda\ k} (e^{\lambda} - 1)\ (1)$

b) for the Bayes theorem is...

$\displaystyle P\{X > k + x| G > k \} = \frac{P \{ X > k + x \}}{P\{X>k \}} = \frac{e^{- \lambda\ (k + x)}}{e^{- \lambda\ k}} = e^{- \lambda\ x}\ (2) $

Kind regards

$\chi$ $\sigma$
 
chisigma said:
a) is...

$\displaystyle P \{ G = k\} = \int_{k-1}^{k} e^{- \lambda\ x}\ d x = e^{\lambda\ k} (e^{\lambda} - 1)\ (1)$

b) for the Bayes theorem is...

$\displaystyle P\{X > k + x| G > k \} = \frac{P \{ X > k + x \}}{P\{X>k \}} = \frac{e^{- \lambda\ (k + x)}}{e^{- \lambda\ k}} = e^{- \lambda\ x}\ (2) $

Kind regards

$\chi$ $\sigma$
Lol... once again, a simple step that I did not see.

Thank you
 
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