Final answer:
The cumulative distribution function (CDF) of an exponential distribution with rate parameter m can be verified by integrating the probability density function (PDF), resulting in the formula P(X ≤ x) = 1 - e-mx, which confirms the memoryless property of the distribution.
Step-by-step explanation:
If X is an exponential distribution with parameter m, we can verify the cumulative distribution function (CDF) formula. By definition, the probability density function (PDF) of an exponential distribution for a continuous random variable X with parameter m is f(x) = me-mx, where x ≥ 0 and m > 0.
The CDF is the probability that X takes on a value less than or equal to x, expressed as P(X ≤ x). It is found by integrating the PDF:
P(X ≤ x) = ∫ f(t) dt from 0 to x = ∫ me-mt dt = -e-mt | from 0 to x = 1 - e-mx.
This resulting function gives us the probability that X is less than or equal to some value x, and is represented as 1 - e-mx. This CDF confirms that the distribution is memoryless, meaning P(X > x + k|X > x) = P(X > k).