Derive or verify variance formula

4. The continuous random variable \(X\) is uniformly distributed over the interval \([ - 4,6 ]\).

1. Write down the mean of \(X\).
2. Find \(\mathrm { P } ( X \leqslant 2.4 )\)
3. Find \(\mathrm { P } ( - 3 < X - 5 < 3 )\) The continuous random variable \(Y\) is uniformly distributed over the interval \([ a , 4 a ]\).
4. Use integration to show that \(\mathrm { E } \left( Y ^ { 2 } \right) = 7 a ^ { 2 }\)
5. Find \(\operatorname { Var } ( Y )\).
6. Given that \(\mathrm { P } \left( X < \frac { 8 } { 3 } \right) = \mathrm { P } \left( Y < \frac { 8 } { 3 } \right)\), find the value of \(a\).

3

1. The continuous random variable \(T\) follows a rectangular distribution with probability density function given by $$\mathrm { f } ( t ) = \left\{ \begin{array} { l c } k & - a \leqslant t \leqslant b \\ 0 & \text { otherwise } \end{array} \right.$$
   1. Express \(k\) in terms of \(a\) and \(b\).
   2. Prove, using integration, that \(\mathrm { E } ( T ) = \frac { 1 } { 2 } ( b - a )\).
2. The error, in minutes, made by a commuter when estimating the journey time by train into London may be modelled by the random variable \(T\) with probability density function $$\mathrm { f } ( t ) = \left\{ \begin{array} { c c } \frac { 1 } { 10 } & - 4 \leqslant t \leqslant 6 \\ 0 & \text { otherwise } \end{array} \right.$$
   1. Write down the value of \(\mathrm { E } ( T )\).
   2. Calculate \(\mathrm { P } ( T < - 3\) or \(T > 3 )\).

5

1. The continuous random variable \(X\) follows a rectangular distribution with probability density function defined by $$f ( x ) = \begin{cases} \frac { 1 } { b } & 0 \leqslant x \leqslant b \\ 0 & \text { otherwise } \end{cases}$$
   1. Write down \(\mathrm { E } ( X )\).
   2. Prove, using integration, that $$\operatorname { Var } ( X ) = \frac { 1 } { 12 } b ^ { 2 }$$
2. At an athletics meeting, the error, in seconds, made in recording the time taken to complete the 10000 metres race may be modelled by the random variable \(T\), having the probability density function $$f ( t ) = \left\{ \begin{array} { c c } 5 & - 0.1 \leqslant t \leqslant 0.1 \\ 0 & \text { otherwise } \end{array} \right.$$ Calculate \(\mathrm { P } ( | T | > 0.02 )\).

4 A continuous random variable \(X\) has a probability density function defined by $$f ( x ) = \begin{cases} \frac { 1 } { k } & a \leqslant x \leqslant b \\ 0 & \text { otherwise } \end{cases}$$ where \(b > a > 0\).

1. 1. Prove that \(k = b - a\).
   2. Write down the value of \(\mathrm { E } ( X )\).
   3. Show, by integration, that \(\mathrm { E } \left( X ^ { 2 } \right) = \frac { 1 } { 3 } \left( b ^ { 2 } + a b + a ^ { 2 } \right)\).
   4. Hence derive a simplified formula for \(\operatorname { Var } ( X )\).
2. Given that \(a = 4\) and \(\operatorname { Var } ( X ) = 3\), find the numerical value of \(\mathrm { E } ( X )\).
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3 The continuous random variable \(R\) follows a rectangular distribution with probability density function given by $$f ( r ) = \begin{cases} k & - a \leq r \leq b \\ 0 & \text { otherwise } \end{cases}$$ Prove, using integration, that \(\mathrm { E } ( R ) = \frac { 1 } { 2 } ( b - a )\) [0pt] [4 marks]

In a computer game, a star moves across the screen, with constant speed, taking 1 s to travel from one side to the other. The player can stop the star by pressing a key. The object of the game is to stop the star in the middle of the screen by pressing the key exactly 0.5 s after the star first appears. Given that the player actually presses the key 7 s after the star first appears, a simple model of the game assumes that T is a continuous uniform random variable defined over the interval [0, 1].

1. Write down P(T < 0.2). [1]
2. Write down E(T). [1]
3. Use integration to find Var(T). [4]

A group of 20 children each play this game once.

1. Find the probability that no more than 4 children stop the star in less than 0.2 s. [3]

The children are allowed to practise this game so that this continuous uniform model is no longer applicable.

1. Explain how you would expect the mean and variance of T to change. [2]

It is found that a more appropriate model of the game when played by experienced players assumes that T has a probability density function g(t) given by $$g(t) = \begin{cases} 4t, & 0 \leq t \leq 0.5, \\ 4 - 4t, & 0.5 \leq t \leq 1, \\ 0, & otherwise. \end{cases}$$

1. Using this model show that P(T < 0.2) = 0.08. [2]

A group of 75 experienced players each played this game once.

1. Using a suitable approximation, find the probability that more than 7 of them stop the star in less than 0.2 s. [4]

The random variable \(X\) follows a continuous uniform distribution over the interval \([2, 11]\).

1. Write down the mean of \(X\). [1 mark]
2. Find P(\(X \geq 8.6\)). [2 marks]
3. Find P(\(|X - 5| < 2\)). [2 marks]

The random variable \(Y\) follows a continuous uniform distribution over the interval \([a, b]\).

1. Show by integration that $$\text{E}(Y^2) = \frac{1}{3}(b^2 + ab + a^2).$$ [5 marks]
2. Hence, prove that $$\text{Var}(Y) = \frac{1}{12}(b - a)^2.$$ You may assume that E(\(Y\)) = \(\frac{1}{2}(a + b)\). [4 marks]