SmarterEd

Aussie Maths & Science Teachers: Save your time with SmarterEd

Statistics, 2ADV S3 2023 HSC 29

A continuous random variable has probability density function given by
 

 

  1. Find the mode of .  (2 marks)

  2. Find the cumulative distribution function for the given probability density function.  (2 marks)

  3. Without calculating the median, show that the mode is greater than the median.  (2 marks)

Show Answers Only

a.   

b.   

c.   

Show Worked Solution

a.    

♦ Mean mark (a) 45%.


  

b.    
   

 


 

 
c.   

 
   

 

♦♦♦ Mean mark (c) 17%.

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)

Show Answers Only
Show Worked Solution
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)

Show Answers Only

Show Worked Solution

 

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)

Show Answers Only
Show Worked Solution
a.  
 
 
 
 
 

 

b.