Final answer:
For x following an exponential distribution with a mean of 10, m, the decay parameter, is 0.1, and the probability density function is f(x) = 0.1e^-0.1x for x > 0. The standard deviation is also 10, and this distribution is memoryless, meaning past behavior doesn't change future probabilities.
Step-by-step explanation:
When x has an exponential distribution with a mean of 10, this means we're dealing with a continuous probability distribution often used to model the time between events in a Poisson process. The parameter m represents the decay rate, which can also be thought of as the rate parameter of the distribution. To find m, we use the relationship that the mean (μ) of an exponential distribution is the reciprocal of the decay parameter, so
m = 1/μ. In this case, with a mean of 10, m would be 0.1. The probability density function (pdf) would be f(x) = 0.1e-0.1x where x > 0.
The standard deviation of an exponential distribution is equal to its mean, so in this case, it would also be 10. Exponential distributions are characterized by the fact that events occur independently and the probability of occurrence is proportional to the length of the interval. Since the distribution is memoryless, the past behavior doesn't affect future probabilities. For example, the probability of waiting another 10 units of time till the next event is the same, regardless of how long you've already waited, as long as we're talking about time intervals beyond a certain point.