# 09_20 - 0 ≤ x ≤ 4 f x = 0 otherwise a What must the...

This preview shows pages 1–4. Sign up to view the full content.

STAT 400 Examples for 09/20/2011 Fall 2011 Continuous Random Variables. The probabilities associated with a continuous random variable X are determined by the probability density function of the random variable. The function, denoted ƒ ( x ), must satisfy the following properties: 1. ƒ ( x ) 0 for all x . 2. The total area under the entire curve of ƒ ( x ) is equal to 1.00. Then the probability that X will be between two numbers a and b is equal to the area under ƒ ( x ) between a and b . For any point c , P(X = c) = 0. Therefore, P(a X b) = P(a X < b) = P(a < X b) = P(a < X < b). Expected value (mean, average): - = x x x d f ) ( X μ . Variance: σ X 2 = ( ) [ ] ( ) - - = - dx x x f ) ( X E 2 X 2 X μ μ . σ X 2 = ( ) ( ) [ ] ( ) 2 X 2 2 2 μ ) ( X E X E - = - - dx x x f . Moment Generating Function: M X ( t ) = E ( e t X ) = ( ) - dx x f x t e .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
1. Let X be a continuous random variable with the probability density function f ( x ) = k x , 0 x 4, f ( x ) = 0,

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 0 ≤ x ≤ 4, f ( x ) = 0, otherwise. a) What must the value of k be so that f ( x ) is a probability density function? b) Find the cumulative distribution function of X, F X ( x ) = P ( X ≤ x ). c) Find the probability P ( 1 ≤ X ≤ 2 ). d) Find the median of the distribution of X. That is, find m such that P ( X ≤ m ) = P ( X ≥ m ) = 1 / 2 . e) Find the 30th percentile of the distribution of X. That is, find a such that P ( X ≤ a ) = 0.30. f) Find μ X = E ( X ). g) Find σ X = SD ( X ). 2. Let X be a continuous random variable with the cumulative distribution function F ( x ) = 0, x < 0, F ( x ) = 8 3 ⋅ x , 0 ≤ x ≤ 2, F ( x ) = 1 – 2 1 x , x > 2. a) Find the probability density function f ( x ). b) Find the probability P ( 1 ≤ X ≤ 4 ). c) Find μ X = E ( X ). d) Find σ X = SD ( X )....
View Full Document

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern