Cumulative Distribution Function Cumulative Distribution Function Distribution Function

Cumulative Distribution Function

Cumulative Distribution Function ( Distribution Function ) The cumulative distribution function , F(x ) , for a continuous random variable X with its probability density function f(t ) valid in the interval [ a , b ] , is defined as f(t) F(x)=P(X≤x) F( x ) If f( t ) valid for ( - ∞ < t < ∞ ) , then F(x) a 0 x bt Example 1 : The continuous random variable X has a probability density function f(x) where f(x) = , 0, 1 ≤ x ≤ 3, otherwise. ( a ) Find the cumulative distribution function for X. Ex ( b ) Find P( X < 2 ) and P( X ≥ 1 ). ( c ) Sketch the graph for f(x) and F(x). 8. 2 pg. 361 Q. 1 – 4

Example 2 : The continuous random variable X has a probability density function f(x) where f(x) = , 1 ≤ x ≤ 2, , 4 ≤ x ≤ 6, , 9 ≤ x ≤ 10, (b) P (1. 5 < X ≤ 4. 5 ) = F ( 4. 5 ) – F ( 1. 5 ) otherwise. = 0. 225 ( a ) Find the cumulative distribution function for X. ( b ) Hence, find P ( 1. 5 < X ≤ 4. 5 ) Ex 8. 2 pg. 361 f(t) — F( x 1 ) f(t) P (x 1 F( ≤ Xx 2≤ )x 2 ) = F ( x 2 ) – F ( x 1 ) 0 a x 1 b 0 a x 2 b P ( a ≤ X ≤ x 1 ) = F ( x 1 ) P ( a ≤ X ≤ x 2 ) = F ( x 2 ) Q. 5 – 7

0. 5 f(x) 0. 5 F( x ) If m = median for X = F( m ) 0 a m b X If probability density function f(x ) is valid in the interval [ a , b ] Then, F( x ) = 1 = F( m ) , where m = median of X F( m ) where m = median of X F( q 1 ) where q 1 = first quartile of X F( q 3 ) where q 3 = third quartile of X

Example 3 : The continuous random variable X has a probability density function f(x) where f(x) = 0, , 0≤x≤ 4, , 10 ≤ x ≤ 14 , otherwise. Find the cumulative distribution function for X. Hence, find the (a) median , m = 11 (b) first quartile , q 1 = 3 (c) third quartile , q 3 =12. 5 Ex 8. 2 pg. 361 Q. 8 – 11

Determining the Probability Density Function from the Cumulative Distribution Function Example 1 : The continuous random variable X has the cumulative distribution function F(x) where 0 F(x) = 1 (a) (b) (c) (d) , x < - 2, , - 2 ≤ x < 0, , 0 ≤ x < 2, , 2 ≤ x < 7, , 7 ≤ x < 15, , x ≥ 15. Find the probability density function for X. Sketch the graph of f (x) and F (x) Find P ( 1. 5 < X ≤ 4. 5 ) Find P ( | X | > 1 ) f ( x ) = F‘ ( x ) Ex 8. 3 pg. 367 Q. 1 – 3 , 8 – 10

Example 2 : The continuous random variable X has the cumulative distribution function 0 F(x) = 1 , x<-1 , -1≤x≤ 0 , 0≤x≤ 1 , x >1 ( a ) Find the values of a and b. ( b ) Find the probability density function for X. ( c ) Find P ( - 5 < 2 X – 2 ≤ - 1 ) Ex 8. 3 pg. 367 Q. 4 – 7

STPM 2004 QUESTION 8 : The continuous random variable X has a probability density function , f(x) = 0, 0<x <3 otherwise. ( i ) Calculate P ( X < [ 3 marks ] ) ( ii ) Find the cumulative distribution function of X. [ 3 marks ] STPM 2004 QUESTION 11 : The discrete random variable X has a probability function P( X = x ) = k ( 4 – x )2 , 0 < x < 3, 0, otherwise. where k is a constant. ( i ) Determine the value of k and tabulate the probability distribution of X. [ 3 marks ] ( ii ) Find E( 7 X – 1 ) and Var ( 7 X – 1 ) [ 7 marks ]