A data scientist is analyzing the response times of a critical API to ensure it meets its Service Level Agreement (SLA). The SLA requires that 95% of all requests have a response time of 200ms or less. After modeling the response time data with a continuous random variable, the scientist needs to select a function that can most directly answer two key questions:
What is the probability that a random request will have a response time less than or equal to 200ms?
What is the response time (in ms) at which 95% of requests are faster (i.e., the 95th percentile)?
Which of the following functions is best suited for this analysis?
The correct answer is the Cumulative Distribution Function (CDF). The CDF of a random variable X, denoted as F(x), is defined as the probability that X will take a value less than or equal to x (F(x) = P(X ≤ x)). This directly answers the first question by evaluating F(200). The second question asks for the 95th percentile, which is the value 'x' for which F(x) = 0.95. This is found using the inverse of the CDF, also known as the quantile function.
The Probability Density Function (PDF) is incorrect because for a continuous variable, it gives the relative likelihood at a specific point, not a cumulative probability. To find the probability of being less than or equal to 200ms, one would need to integrate the PDF from its minimum value up to 200.
The Probability Mass Function (PMF) is incorrect because it applies to discrete random variables, whereas response time is a continuous variable. Like a PDF, a PMF gives the probability at a specific point, not a cumulative value.
The Survival Function, S(x), gives the probability that a variable takes a value greater than x (S(x) = P(X > x)). It is the complement of the CDF (S(x) = 1 - F(x)). While it could be used to solve the problem indirectly, the CDF is the function that most directly answers the questions as posed.
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