Final answer:
The cumulative distribution function (CDF) represents the probability that a random variable is less than or equal to a specific value. For a logistic distribution, the CDF has an S-shaped curve and can be used to solve for particular probabilities, such as finding the value of x that corresponds to a given probability by using logarithms.
Step-by-step explanation:
Logistic Distribution and Cumulative Distribution Function (CDF)
The cumulative distribution function, or CDF, of a random variable X, is denoted by F(x) and it represents the probability that X is less than or equal to a particular value x. Specifically, the CDF is given by the equation F(x) = P(X ≤ x). In the context of the student's question, a logistic distribution is being discussed. This distribution has a characteristic S-shaped curve, known as the logistic curve, which can be mathematically represented by a specific cumulative distribution function.
The general form of a CDF for a logistic distribution is F(x) = (1 + e-kx)-1, where k is a parameter that affects the steepness of the curve. This logistic curve is often compared to an exponential curve, which would result without any negative feedback. When dealing with continuous probability distributions, the total area under the curve (and above the x-axis) of the probability density function (pdf) is equal to one, indicating all possible outcomes.
To solve for specific values, one might set a certain probability, such as 0.70, equal to the CDF, and solve for x. For instance, if the CDF is P(X < x) = 1 - e-0.5x, and we want to solve for the x that satisfies P(X < x) = 0.70, we would end up with the equation 0.70 = 1 - e-0.5x, allowing us to solve for x using logarithms to find the desired quantile of the distribution.
The complete question is: The logistic distribution is associated with the cdf F(x) = (1 +e-*)-1, - is: