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When I was a wee lad just out of school 3 decades ago, I witnessed an event more astounding than breaking the sound barrier in aviation, namely, breaking the unity barrier in probability. An analyst with a Ph.D. in Physics had spent 2 years full-time (probably giving 150%) developing a model for calculating the probability of detecting object X, at the end of which his model and analysis successfully completed peer review by several scientists and engineers closely affiliated to the U.S. government. I won't tell you what object X is, but object X, and the probability of detecting it, was and still is of considerable interest to the U.S. government. The model included a formula for P_y = Prob(event y happens). P_y and some other terms all combined into the final formula, which was Prob(object X is detected). Indeed, computed values of Prob(object X is detected) were within the range of [0,1], as is "traditional" in probability in the Kolmogorov tradition. P_y in its original form was always in [0,1] and involved "garden-variety" transcendental functions which were available in standard Fortran or any scientific calculator. However, for a reason known only but to the analyst and God (perhaps because he had seen it done in his Physics classes and books, but did not know that he was shown the few cases where it works, not the many more where it does not, and this guy's name and scientific/mathematical judgment did not happen to be that of Dirac), he chose to take a two term Taylor expansion of P_y (and ignore the remainder term), which will henceforth be referred to as P_y. It was this two term Taylor expansion of P_y which was inserted into the final expression for Prob(object X is detected). What he did not realize, until I pointed it out to him, was that P_y was equal to approximately 1.2 using his base case values for all parameters. Indeed it was possible for P_y to go up to about 1.8. And that's how the unity barrier was broken in probability. But the guy didn't know he had accomplished this pioneering feat until I pointed it out to him, having just performed quick calculations on a battery-powered credit card size Casio scientific calculator in a darkened conference room (couldn't have done it with a solar-powered calculator). That would be kind of like Chuck Yeager going out for a Sunday spin in his plane, and only being informed months later that he had broken the sound barrier.

When I was a wee lad just out of school 3 decades ago, I witnessed an event more astounding than breaking the sound barrier in aviation, namely, breaking the unity barrier in probability. An analyst with a Ph.D. in Physics had spent 2 years full-time (probably giving 150%) developing a model for calculating the probability of detecting object X, at the end of which his model and analysis successfully completed peer review by several scientists and engineers closely affiliated to the U.S. government. I won't tell you what object X is, but object X, and the probability of detecting it, was and still is of considerable interest to the U.S. government. The model included a formula for $P_y$ = Prob(event y happens). $P_y$ and some other terms all combined into the final formula, which was Prob(object X is detected). Indeed, computed values of Prob(object X is detected) were within the range of [0,1], as is "traditional" in probability in the Kolmogorov tradition. $P_y$ in its original form was always in [0,1] and involved "garden-variety" transcendental functions which were available in standard Fortran or any scientific calculator. However, for a reason known only but to the analyst and God (perhaps because he had seen it done in his Physics classes and books, but did not know that he was shown the few cases where it works, not the many more where it does not, and this guy's name and scientific/mathematical judgment did not happen to be that of Dirac), he chose to take a two term Taylor expansion of $P_y$ (and ignore the remainder term), which will henceforth be referred to as $P_y$. It was this two term Taylor expansion of $P_y$ which was inserted into the final expression for Prob(object X is detected). What he did not realize, until I pointed it out to him, was that $P_y$ was equal to approximately 1.2 using his base case values for all parameters. Indeed it was possible for $P_y$ to go up to about 1.8. And that's how the unity barrier was broken in probability. But the guy didn't know he had accomplished this pioneering feat until I pointed it out to him, having just performed quick calculations on a battery-powered credit card size Casio scientific calculator in a darkened conference room (couldn't have done it with a solar-powered calculator). That would be kind of like Chuck Yeager going out for a Sunday spin in his plane, and only being informed months later that he had broken the sound barrier.

Back to the original question "Can a probability distribution value exceeding 1 be OK?" NO, but I've seen it done (see my last paragraph below).

Back to the original question "Can a probability distribution value exceeding 1 be OK?" No, but I've seen it done (see my last paragraph below).

When I was a wee lad just out of school 3 decades ago, I witnessed an event more astounding than breaking the sound barrier in aviation, namely, breaking the unity barrier in probability. An analyst with a Ph.D. in Physics had spent 2 years full-time (probably giving 150%) developing a model for calculating the probability of detecting object X, at the end of which his model and analysis successfullly completed peer review by several scientists and engineers closely affiliated to the U.S. government. I won't tell you what object X is, but object X, and the probability of detecting it, was and still is of considerable interest to the U.S. government. The model included a formula for P_y = Prob(event y happens). P_y and some other terms all combined into the final formula, which was Prob(object X is detected). Indeed, computed values of Prob(object X is detected) were within the range of [0,1], as is "traditional" in probability in the Kolmogorov tradition. P_y in its original form was always in [0,1] and involved "garden-variety" transcendental functions which were available in standard Fortran or any scientific calculator. However, for a reason known only but to the analyst and God (perhaps because he had seen it done in his Physics classes and books, but did not know that he was shown the few cases where it works, not the many more where it does not, and this guy's name and scientific/mathematical judgment did not happen to be that of Dirac), he chose to take a two term Taylor expansion of P_y (and ignore the remainder term), which will henceforth be referred to as P_y. It was this two term Taylor expansion of P_y which was inserted into the final expression for Prob(object X is detected). What he did not realize, until I pointed it out to him, was that P_y was equal to approximately 1.2 using his base case values for all parameters. Indeed it was possible for P_y to go up to about 1.8. And that's how the unity barrier was broken in probability. But the guy didn't know he had accomplished this pioneering feat until I pointed it out to him, having just performed quick calculations on a battery-powered credit card size Casio scientific calculator in a darkened conference room (couldn't have done it with with a solar-powered calculator). That would be kind of like Chuck Yeager going out for a Sunday spin in his plane, and only being informed months later that he had broken the sound barrier.

When I was a wee lad just out of school 3 decades ago, I witnessed an event more astounding than breaking the sound barrier in aviation, namely, breaking the unity barrier in probability. An analyst with a Ph.D. in Physics had spent 2 years full-time (probably giving 150%) developing a model for calculating the probability of detecting object X, at the end of which his model and analysis successfully completed peer review by several scientists and engineers closely affiliated to the U.S. government. I won't tell you what object X is, but object X, and the probability of detecting it, was and still is of considerable interest to the U.S. government. The model included a formula for P_y = Prob(event y happens). P_y and some other terms all combined into the final formula, which was Prob(object X is detected). Indeed, computed values of Prob(object X is detected) were within the range of [0,1], as is "traditional" in probability in the Kolmogorov tradition. P_y in its original form was always in [0,1] and involved "garden-variety" transcendental functions which were available in standard Fortran or any scientific calculator. However, for a reason known only but to the analyst and God (perhaps because he had seen it done in his Physics classes and books, but did not know that he was shown the few cases where it works, not the many more where it does not, and this guy's name and scientific/mathematical judgment did not happen to be that of Dirac), he chose to take a two term Taylor expansion of P_y (and ignore the remainder term), which will henceforth be referred to as P_y. It was this two term Taylor expansion of P_y which was inserted into the final expression for Prob(object X is detected). What he did not realize, until I pointed it out to him, was that P_y was equal to approximately 1.2 using his base case values for all parameters. Indeed it was possible for P_y to go up to about 1.8. And that's how the unity barrier was broken in probability. But the guy didn't know he had accomplished this pioneering feat until I pointed it out to him, having just performed quick calculations on a battery-powered credit card size Casio scientific calculator in a darkened conference room (couldn't have done it with a solar-powered calculator). That would be kind of like Chuck Yeager going out for a Sunday spin in his plane, and only being informed months later that he had broken the sound barrier.