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 7 MC
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)
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- Show that the mode of `X` is two hours. (2 marks)
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- Show that `P(X < 2) = log_10 2`. (2 marks)
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- 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)
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Statistics, 2ADV S3 EQ-Bank 6 MC
Statistics, 2ADV S2 SM-Bank 14
A probability density function `f(x)` is given by
`f(x) = {(px(3 - x), \ text(if)\ \ 0 <= x <= 3),(0, \ text(if)\ \ x < 0\ \ text(or if)\ \ x > 3):}`
where `p` is a positive constant.
- Find the value of `p`. (2 marks)
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- Find the mode of `f(x)`. (2 marks)
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Statistics, 2ADV S2 SM-Bank 15
A function `f(x)` is given by
`f(x) = {(3/4(x - 2)(4 - x), \ text(if)\ \ 2 <= x <= 4),(0, \ text(if)\ \ x < 2\ \ text(or if)\ \ x > 4):}`
- Show this curve is a probability density function. (2 marks)
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- Find the mode. (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 `T`, which models the time, `t,` in minutes, that Jennifer spends each day on her homework, has a probability density function `f`, where
`f(t) = {{:(1/625 (t - 20)),(1/625 (70 - t)),(0):}qquad{:(20 <= t < 45),(45 <= t <= 70),(text(elsewhere)):}:}`