{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# 4.1-4.3 - Chapter 4 Continuous Distributions Definition A...

This preview shows pages 1–6. Sign up to view the full content.

Chapter 4 Continuous Distributions Definition: A function f(x) is called the density of a continuous variable X if (i) ; 0 ) ( x f (ii) 1 ) ( = - dx x f (a) The graph of ) ( x f is called “density curve”. (b) Proportion of X - values between ‘a’ and ‘b’=P( b X a ) = ) ( b a dx x f = area of density curve between a and b (c ) Proportion of X - values with b X a = Proportion of x with b X a < < . ( Since the area of the curve under single value = 0, P(X=a)=0 for any constant a). 1

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

View Full Document
Examples of the Graphs of Continuous Distributions: 1. Sketch the graph of uniform distribution and give the formula for the m ass function f(x) a. on the interval [0,1] b. on [a,b] 2. Exponential distribution x 0 >0 ( ) 0 x<0 x e f x λ - = The Graph: Verify that it is a legitimate distribution. (Is ( ) 1 f x f - = ?) How can we find a proportion of all values X that fall between two given numbers? Answer: . ... 2
Example 1 : Let X denotes the amount of time ( hr ) the music is played in a program of a radio station. A potential sponsor wants to study the distribution of x. Suppose, the density function is ) ( x f = 90 x 8 (1- x ), 0 ≤ x 1 0, otherwise. (i) Note ) ( x f is a density. (ii) Proportion of programs between 0.7 and 0.9 (hr) is . 587 . 0 ) ( 9 . 7 . = dx x f (iii) The constant c such that .5 ) ( ) ( 1 = = c c O dx x f dx x f 3

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

View Full Document
.5 10 9 90 10 9 = - c c Solving by a numerical procedure, c 2245 .838 50% of programs have music < .838 hrs. HW assignment: Section 4.1: #1-9 odd, 8* 4
4.2 Cumulative distribution functions and expected Values Cumulative distribution(cdf): Let f(x) be the density of a random variable X then F(x)=P(X x)= - x dt t f ) ( is called the cumulative distribution function of X. Example. If f(x)=2x, 0<x<1, f(x)=0 otherwise then F(x)= - x dt t f ) ( = 2 0 2 x tdt x = for 0<x<1, F(x)=1, for x 1 and F(x)=0, for x . 0 Using F(x) to calculate probabilities. We can use F to calculate probabilities, P(a X b )=F(b)-F(a). P(X>0.5)=1-F(0.5)……

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.

{[ snackBarMessage ]}

### Page1 / 17

4.1-4.3 - Chapter 4 Continuous Distributions Definition A...

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online