A continuous random variable \(X\) has probability density function \(f(x)\) given by \(f(x)=\left\{\begin{array}{cl} 12 x^2(1-x), & \text { for } 0 \leq x \leq 1 \\ 0, & \text { for all other values of } x \end{array}\right.\) --- 6 WORK AREA LINES (style=lined) --- --- 6 WORK AREA LINES (style=lined) --- --- 5 WORK AREA LINES (style=lined) ---
Statistics, 2ADV S3 2022 HSC 30
A continuous random variable `X` has cumulative distribution function given by
`F(x)={[1,x > e^(3)],[(1)/(k)ln x,1 <= x <= e^(3)],[0,x < 1]:}` .
- Show that `k = 3`. (1 mark)
--- 2 WORK AREA LINES (style=lined) ---
- Given that `P(X < c)=2P(X > c)`, find the exact value of `c`. (2 marks)
--- 4 WORK AREA LINES (style=lined) ---
Statistics, 2ADV S3 2021 HSC 33
People are given a maximum of six hours to complete a puzzle. The time spent on the puzzle, in hours, can be modelled using the continuous random variable `X` which has probability density function
`f(x) = {{:((Ax)/(x^2 +\ 4),),(0,):}{:(text(for)\ 0 <= x <= 6 text{, (where}\ A >\ text{0)}),(text(for all other values of)\ \ x):}:}`
The graph of the probability density function is shown below. The graph has a local maximum.
- Show that `A = 2/(ln 10)`. (2 marks)
--- 4 WORK AREA LINES (style=lined) ---
- Show that the mode of `X` is two hours. (2 marks)
--- 4 WORK AREA LINES (style=lined) ---
- Show that `P(X < 2) = log_10 2`. (2 marks)
--- 5 WORK AREA LINES (style=lined) ---
- The Intelligence Quotient (IQ) scores of people are normally distributed with a mean of 100 and standard deviation of 15.
- It has been observed that the puzzle is generally completed more quickly by people with a high IQ.
- It is known that 80% of people with an IQ greater than 130 can complete the puzzle in less than two hours.
- A person chosen at random can complete the puzzle in less than two hours.
- What is the probability that this person has an IQ greater than 130? Give your answer correct to three decimal places. (2 marks)
--- 4 WORK AREA LINES (style=lined) ---
Statistics, 2ADV S3 2021 HSC 30
The number of hours for which light bulbs will work before failing can be modelled by the random variable `X` with cumulative distribution function
`F(x) = { {:(1 - e^(-0.01x)),(0):} {: (\ \ x >= 0), (\ \ x < 0):} :}`
Jane sells light bulbs and promises that they will work for longer than exactly 99% of all light bulbs.
Find how long, according to Jane’s promise, a light bulb bought from her should work. Give your answer in hours, rounded to two decimal places. (2 marks)
--- 5 WORK AREA LINES (style=lined) ---
Statistics, 2ADV S3 EQ-Bank 1
A probability density function can be used to model the lifespan of a termite, `X`, in weeks, is given by
`f(x) = {(k(36 - x^2)),(0):}\ \ \ {:(3 <= x <= 6),(text(otherwise)):}`
- Show that the value of `k` is `1/45`. (2 marks)
--- 4 WORK AREA LINES (style=lined) ---
- Find the cumulative distribution function. (2 marks)
--- 4 WORK AREA LINES (style=lined) ---
- Find the probability that a termite's lifespan is greater than 5 weeks. (1 mark)
--- 2 WORK AREA LINES (style=lined) ---
Statistics, 2ADV S3 EQ-Bank 16
A probability density function is defined by
`f(x) = {(a, \ text(for)\ \ 0 <= x <= 4),(3a, \ text(for)\ 4 < x <= 8):}`
- Find the value of `a`. (2 marks)
--- 4 WORK AREA LINES (style=lined) ---
- Sketch the probability density function. (1 mark)
--- 6 WORK AREA LINES (style=lined) ---
- Find an expression for the cumulative distribution function. (2 marks)
--- 5 WORK AREA LINES (style=lined) ---