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Statistics, 2ADV S3 2020 HSC 23

A continuous random variable, , has the following probability density function.
 


 

  1. Find the value of .  (2 marks)

  2. Find  . Give your answer correct to four decimal places.  (2 marks)

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a.   
 
 
 
 
 

 

♦ Mean mark part (b) 44%.
b.   
   
   
   
   
   

Statistics, 2ADV S3 EQ-Bank 1

A probability density function can be used to model the lifespan of a termite, , in weeks, is given by
 


 

  1. Show that the value of    is  .  (2 marks)

  2. Find the cumulative distribution function.  (2 marks)

  3. Find the probability that a termite's lifespan is greater than 5 weeks.  (1 mark)

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i.   
 
 
 
 

 

ii.   
   
   
   
   
   
   

 

 

iii.   
   
   
   

Statistics, 2ADV S3 SM-Bank 18

The Lorenz birdwing is the largest butterfly in a habitat.

The probability density function that describes its life span, , in weeks, is given by
 


 

In a sample of 80 Lorenz birdwing butterflies, how many butterflies are expected to live longer than two weeks, correct to the nearest integer?  (2 marks)

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Statistics, 2ADV S3 SM-Bank 13

The time Jennifer spends on her homework each day varies, but she does some homework every day.

The continuous random variable , which models the time, , in minutes, that Jennifer spends each day on her homework, has a probability density function , where

  1. Sketch the graph of  on the axes provided below.  (3 marks)

     

        
     

  2. Find the mode.  (1 mark)

  3. Find  .  (2 marks)

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  1.  

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i.   

MARKER’S COMMENT: Many did not draw graph along -axis between 0 and 20 and for  .

 
ii.     
 

iii.    

Statistics, 2ADV S3 SM-Bank 8

The continuous random variable has a distribution with probability density function given by
 


 

where is a positive constant.

  1. Find the value of  .  (3 marks)

  2. Express    as a  definite integral. (Do not evaluate the definite integral.)  (1 mark)

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a.  
 
 
 
 
 

 

b.