Final answer:
The mean and standard deviation of the exponential probability density function for radioactive particle decay time are both 0.1, and the probability of the variable being between the mean and one standard deviation above the mean is approximately 0.6321, which is not exactly represented by any of the options.
Step-by-step explanation:
The question is about finding the mean, standard deviation, and the probability between the mean and one standard deviation above the mean for a radioactive particle decay time, modeled by an exponential distribution with a probability density function (PDF). The PDF given is f(t) = 10e^(-10t) for t ≥ 0, which describes the time until a radioactive particle decays.
The mean (or expected value) of an exponential distribution is the reciprocal of the rate parameter, which in this case is 1/10, giving us a mean of 0.1. To find the standard deviation, we use the fact that for an exponential distribution, the standard deviation is equal to the inverse of the rate parameter as well, so the standard deviation is also 0.1.
Finally, for an exponential distribution, the probability that the random variable is between the mean (0.1) and one standard deviation above the mean (0.2) can be found by integrating the PDF from 0.1 to 0.2, but an easier way is to use the properties of the exponential distribution to state that the probability of being within one standard deviation of the mean is approximately 0.6321 (1 - e^(-1)). Therefore, the answer is closest to option A.
Option A) Mean: (0.1), Standard Deviation: (0.1), Probability: (0.3413) - This is nearly, but not exactly correct (the probability number is rounded off and not correct for an exponential distribution).