Question

Consider the following exponential probability density function. f(x) = e⁻ˣ/⁴ for x≥0. Which is the formula for P(x≤x0)?

1) P(x≤x0) = 1 - e⁻ˣ₀/⁴
2) P(x≤x0) = e⁻ˣ₀/⁴
3) P(x≤x0) = 1 - e⁻⁴/ˣ₀
4) P(x≤x0) = e⁻⁴/ˣ₀

Answer 1

The correct formula for P(x≤x0) for the given exponential probability density function f(x) = e⁻¹⁴ for x≥0 is P(x≤x0) = 1 - e⁻¹⁴⁰0/4.

Step-by-step explanation:

The student is asking about the cumulative distribution function (CDF) for an exponential probability density function. Specifically, the function is given as f(x) = e⁻¹⁴ for x≥0. To find P(x≤x0), which is the probability that the random variable X is less than or equal to a certain value x0, we need to integrate the probability density function from 0 to x0. Doing so, we obtain the CDF, which, in the case of an exponential distribution, is 1 - e⁻⁴¹⁴. Therefore, the correct formula for P(x≤x0) is P(x≤x0) = 1 - e⁻¹⁴⁰0/4, which corresponds to option 1 in the question provided.