Continuous Random Variables: Probability Density Functions (University)
Free printable university probability worksheet on continuous random variables: verify a pdf integrates to 1, find the normalising constant k, and explain why P(X = a) = 0. Every integral is evaluated exactly from the closed-form antiderivative.
✓ Answer key checked by math, never wrong
Probability & Statistics · Math worksheet
Name
Date
Math
Continuous Random Variables: Probability Density Functions
A function f is a valid pdf when f(x) >= 0 everywhere and its integral over the support is exactly 1. Verify each pdf, find the normalising constant, or evaluate the point probability.
- 1.X is a continuous random variable with pdf f(x) = 3x^2125 for 0 <= x <= 5, and f(x) = 0 otherwise. State P(X = 5) exactly, and explain why.
- 2.Find the constant k so that f(x) = kx2 for 0 <= x <= 2 (and f(x) = 0 otherwise) is a valid probability density function.
- 3.Show that f(x) = 2x/25 for 0 <= x <= 5, and f(x) = 0 otherwise is a valid probability density function by checking f(x) >= 0 and integrating over its support.
- 4.Find the constant k so that f(x) = k for 2 <= x <= 5 (and f(x) = 0 otherwise) is a valid probability density function.
- 5.Show that f(x) = 3x^264 for 0 <= x <= 4, and f(x) = 0 otherwise is a valid probability density function by checking f(x) >= 0 and integrating over its support.
- 6.Find the constant k so that f(x) = kx for 0 <= x <= 3 (and f(x) = 0 otherwise) is a valid probability density function.
- 7.X is a continuous random variable with pdf f(x) = 2x/16 for 0 <= x <= 4, and f(x) = 0 otherwise. State P(X = 2) exactly, and explain why.
- 8.Is f(x) = 69(3 - x) for 0 <= x <= 3 (and f(x) = 0 otherwise) a valid probability density function? Justify by integration.
Made with ChalkBee · chalkbee.com