This question is related to Comparing two exponential random variables except that I am interested in calculating the probability that $A = B$ and given $\lambda_a = \lambda_b = \lambda$. I tried calculating it as $\int_0^\inf f(x).f(x) dx$ where $f(x)$ is pdf of exponentially distributed variable, which gives me answer as $\lambda / 2$ but this does not pass the smell test for me. I would expect answer to be independent of $\lambda$ since $\lambda$ is not dimensionless whereas the answer should be dimensionless.
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Were you expecting the answer $0$? – Henry Jul 31 '19 at 23:26
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If a real-value random variable has a continuous distribution with no atoms (discrete values with non-zero probabilities), then the probability that two samples will be equal is $0$. – Brian Tung Jul 31 '19 at 23:36
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Ah. I see. I think you are correct. – morpheus Jul 31 '19 at 23:40
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However, your result does tell you that the probability that the two values are within some small range $\varepsilon$ of each other is approximately equal to $\frac\lambda2 \varepsilon$. – Brian Tung Jul 31 '19 at 23:40
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Thanks. I forgot my fundamentals – morpheus Jul 31 '19 at 23:42