Expectation of function of X

A question is this type if and only if it asks to find E(g(X)) for some function g, such as E(√X), E(1/X), or E(X²), by integrating g(x)·f(x).

6 questions

CAIE Further Paper 4 2021 June Q3
3 The continuous random variable \(X\) has cumulative distribution function F given by $$F ( x ) = \begin{cases} 0 & x < 0
\frac { 1 } { 81 } x ^ { 2 } & 0 \leqslant x \leqslant 9
1 & x > 9 \end{cases}$$
  1. Find \(\mathrm { E } ( \sqrt { X } )\).
  2. Find \(\operatorname { Var } ( \sqrt { X } )\).
  3. The random variable \(Y\) is given by \(Y ^ { 3 } = X\). Find the probability density function of \(Y\).
CAIE Further Paper 4 2022 June Q4
4 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} \frac { 3 } { 8 } \left( 1 + \frac { 1 } { x ^ { 2 } } \right) & 1 \leqslant x \leqslant 3
0 & \text { otherwise } \end{cases}$$
  1. Find \(\mathrm { E } ( \sqrt { X } )\).
    The random variable \(Y\) is given by \(Y = X ^ { 2 }\).
  2. Find the probability density function of \(Y\).
  3. Find the 40th percentile of \(Y\).
Edexcel S2 2016 June Q7
7. The weight, \(X \mathrm {~kg}\), of staples in a bin full of paper has probability density function $$f ( x ) = \left\{ \begin{array} { c c } \frac { 9 x - 3 x ^ { 2 } } { 10 } & 0 \leqslant x < 2
0 & \text { otherwise } \end{array} \right.$$ Use integration to find
  1. \(\mathrm { E } ( X )\)
  2. \(\operatorname { Var } ( X )\)
  3. \(\mathrm { P } ( X > 1.5 )\) Peter raises money by collecting paper and selling it for recycling. A bin full of paper is sold for \(\pounds 50\) but if the weight of the staples exceeds 1.5 kg it sells for \(\pounds 25\)
  4. Find the expected amount of money Peter raises per bin full of paper. Peter could remove all the staples before the paper is sold but the time taken to remove the staples means that Peter will have \(20 \%\) fewer bins full of paper to sell.
  5. Decide whether or not Peter should remove all the staples before selling the bins full of paper. Give a reason for your answer.
    \href{http://PhysicsAndMathsTutor.com}{PhysicsAndMathsTutor.com}
WJEC Further Unit 2 2019 June Q4
4. The continuous random variable, \(X\), has the following probability density function $$f ( x ) = \begin{cases} k x & \text { for } 0 \leqslant x < 1
k x ^ { 3 } & \text { for } 1 \leqslant x \leqslant 2
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
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  1. Show that \(k = \frac { 4 } { 17 }\).
  2. Determine \(\mathrm { E } ( X )\).
  3. Calculate \(\mathrm { E } ( 3 X - 1 )\) and \(\operatorname { Var } ( 3 X - 1 )\).
WJEC Further Unit 2 2023 June Q1
  1. The random variable \(X\) has mean 17 and variance 64 . The independent random variable \(Y\) has mean 10 and variance 16 . Find the value of
    1. \(\mathrm { E } ( 4 Y - 2 X + 1 )\),
    2. \(\quad \operatorname { Var } ( 4 Y - 5 X + 3 )\),
    3. \(\mathrm { E } \left( X ^ { 2 } Y \right)\).
    4. For a set of 30 pairs of observations of the variables \(x\) and \(y\), it is known that \(\sum x = 420\) and \(\sum y = 240\). The least squares regression line of \(y\) on \(x\) passes through the point with coordinates \(( 19,20 )\).
    5. Show that the equation of the regression line of \(y\) on \(x\) is \(y = 2 \cdot 4 x - 25 \cdot 6\) and use it to predict the value of \(y\) when \(x = 26\).
    6. State two reasons why your prediction in part (a) may not be reliable.
    7. It is known that the average lifetime of hair dryers from a certain manufacturer is 2 years. The lifetimes are exponentially distributed.
    8. Find the probability that the lifetime of a randomly selected hair dryer is between 1.8 and \(2 \cdot 5\) years.
    9. Given that \(20 \%\) of hair dryers have a lifetime of at least \(k\) years, find the value of \(k\).
    10. Jon buys his first hair dryer from the manufacturer today. He will replace his hair dryer with another from the same manufacturer immediately when it stops working. Find the probability that, in the next 5 years, Jon will have to replace more than 3 hair dryers.
    11. State one assumption that you have made in part (c).
    12. A continuous random variable \(X\) has cumulative distribution function \(F\) given by
    $$F ( x ) = \begin{cases} 0 & \text { for } x < 0
    \frac { 1 } { 4 } x & \text { for } 0 \leqslant x \leqslant 2
    \frac { 1 } { 480 } x ^ { 4 } + \frac { 7 } { 15 } & \text { for } 2 < x \leqslant b
    1 & \text { for } x > b \end{cases}$$
  2. Show that \(b = 4\).
  3. Find \(\mathrm { P } ( X \leqslant 2 \cdot 5 )\).
  4. Write down the value of the lower quartile of \(X\).
  5. Find the value of the upper quartile of \(X\).
  6. Find, correct to three significant figures, the value of \(k\) that satisfies the equation \(\mathrm { P } ( X > 3 \cdot 5 ) = \mathrm { P } ( X < k )\).
AQA Further AS Paper 2 Statistics 2019 June Q4
6 marks
4
  1. \(\text { Find } \mathrm { P } ( X > 1 )\)
    [0pt] [3 marks]
    4

  2. [0pt] [3 marks]
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    4
  3. Find \(\mathrm { E } \left( 2 X ^ { - 1 } - 3 \right)\)