AK7_230

# AK7_230 - ECO 230 Marianna Kudlyak Economic Statistics...

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

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

View Full Document

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: ECO 230 Marianna Kudlyak Economic Statistics Spring 2007 Answer Key to Homework #7 Total points: 10. Graded exercises: Total points: 10. Graded exercises: Exercise 2 (1.5 points),3 (1.5 points), 7(2 points), and 8(3 points). Half point was added for the presence of each of the remaining 4 exercises. Exercise 1 Consider the following probability distribution function: f(x) = 1/c for -3 < x < 7 = 0 otherwise a) Find c so that f(x) is a proper probability distribution function and draw the probability distribution function. f(x) is uniform pdf on the interval (a,b). For f(x) to be proper pdf, the following properties should hold: ∫ ∫ ∈ ≥ = ∞ ∞ − v u R v u for dx x f dx x f , ) ( . 2 1 ) ( . 1 Hence, to find c we integrate 1/c on the interval from a to b and equate it to 1; c = 7-(- 3)=10 Since 10> 0 , graph of f(x) lies above x-axis. Thus, we conclude that second property holds. Answer: c = 10 b) Find and draw the cumulative distribution function. F(x ) = 0 if x 0 < -3 F(x ) = 0 if x 0 > 7 10 ) 3 ( 10 1 ) ( − − = = ∫ x dx x F x a if -3 <= x < = 7 -3 7 1/10 x f(x) c) Find E(X). 2 2 7 3 10 * 2 ) 3 ( 7 10 1 ) ( 2 2 7 3 = + − = − − = = ∫ − dx x X E d) Find var(X). ( ) 12 100 10 * 3 ) 3 ( ) 7 ( 2 10 1 )) ( ( ) ( ) var( 3 3 2 7 3 2 2 2 = − − = − = − = ∫ − dx x X E X E X e) Find the median of X. P(X<=m) = 0.5 5 . 10 1 3 = ∫ − dx m After solving for m , we get m = (-3+7)/2=2 f) Find standard deviation of Y, where Y = 2 – 5X 3 25 3 2 10 5 ) ( 5 ) var( 5 ) 5 2 var( ) var( ) ( = = = = − = = X X X Y Y σ σ . Exercise 2. Recall that Normal distribution is to satisfy the following requirements: the graph should be bell shaped curve; mean, median and mode are all equal; mean, median and mode are located at the center of the distribution; it has only one mode, it is symmetric about mean, it is a continuous function; it never touches x-axis; and the area under curve equals one. Let X ~N(50, 64). In the answers below I standardize the random variable by constructing new random variable ) 1 ; ( ~ N Z X Z σ µ − = a) Find the probability that X is greater than 60. P(X>60) = P(Z>1.25) = 1-F(1.25) = 0.106 -3 7 1 x F(x) b) Find the probability that X is greater than 35 and less than 62. P(35<X<62) = P(-1.875<Z<1.5) = F(1.5) – F(-1.875) = F(1.5) – (1 - F(1.875)) = 0.903 c) Find the probability that X is less than 55....
View Full Document

{[ snackBarMessage ]}

### Page1 / 8

AK7_230 - ECO 230 Marianna Kudlyak Economic Statistics...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online