Math Problem Statement
Suppose that f(x) = e −x for x ≥ 0 is a PDF of a distribution. Determine the following: (a) P(X > 1) (b) P(1 < X < 2.5) (c) P(X = 3) (d) P(X < 4) (e) a > 0 such that P(X ≥ a) = 0.10
Solution
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Math Problem Analysis
Mathematical Concepts
Exponential Distribution
Probability Density Function (PDF)
Cumulative Distribution Function (CDF)
Formulas
PDF: f(x) = λe^{-λx}
CDF: F(x) = 1 - e^{-x}
P(X > x) = 1 - F(x)
P(1 < X < 2.5) = F(2.5) - F(1)
P(X = 3) = 0 (since X is continuous)
Solving for a: F(a) = 0.90 implies a = ln(10)
Theorems
Exponential Distribution Theorem
Properties of Continuous Random Variables (P(X = c) = 0)
Suitable Grade Level
Undergraduate level (Introductory Probability/Statistics)
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