# Probability density functions

• Sep 8th 2008, 04:27 PM
One
Probability density functions
Hi could I please get some urgent help with the following?

1) Let X be a continuous random variable with probability density function given by:

fX(x) = 2x^(−3) for x>=1
_______0 otherwise

(a) Show that fX(x) is a valid density function.

(b) Explain why fX(1) = 2 is an acceptable value for the density function when probabilities must be less than 1.

(c) Find the distribution function for X.

(f) Solve the equation FX(x) = 0.5 for x. What importance does this value have?
• Sep 8th 2008, 05:39 PM
mr fantastic
Quote:

Originally Posted by One
Hi could I please get some urgent help with the following?

1) Let X be a continuous random variable with probability density function given by:

fX(x) = 2x^(−3) for x>=1
_______0 otherwise

(a) Show that fX(x) is a valid density function.

Mr F says: Show that $\displaystyle {\color{red}\int_{1}^{\infty} 2x^{-3} \, dx = 1}$ and $\displaystyle {\color{red}2x^{-3} \geq 0}$ for $\displaystyle {\color{red} x \geq 1}$.

(b) Explain why fX(1) = 2 is an acceptable value for the density function when probabilities must be less than 1. Mr F says: $\displaystyle {\color{red} f_X(1) }$ does not represent a probability.

(c) Find the distribution function for X. Mr F says: Use the definition you will have in your class notes or textbook.

(f) Solve the equation FX(x) = 0.5 for x. What importance does this value have? Mr F says: Equate your answer to (c) to 0.5 and solve for x. This value of x is the median of X.

Go back and revise the basic definitions.