5.03c Calculate mean/variance: by integration

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]

6 The random variable \(T\) can take the value \(T = - 2\) or any value in the range \(0 \leq T < 12\) The distribution of \(T\) is given by \(\mathrm { P } ( T = - 2 ) = c , \mathrm { P } ( 0 \leq T \leq t ) = 225 k - k ( 15 - t ) ^ { 2 }\) 6

1. 1. Show that \(1 - c = 216 k\) [0pt] [3 marks] 6
      1. (ii) Given that \(c = 0.1\), find the value of \(\mathrm { E } ( T )\) [0pt] [3 marks]
         6
   2. Show that \(\mathrm { E } ( \sqrt { | T | } ) = \frac { 5 \sqrt { 2 } + 52 \sqrt { 3 } } { 50 }\) [0pt] [3 marks]

1. The discrete random variable \(X\) has probability distribution

where \(q\) and \(r\) are probabilities.

1. Write down, in terms of \(q , \mathrm { P } ( X \leqslant 0 )\)
2. Show that \(\mathrm { E } \left( X ^ { 2 } \right) = \frac { 7 } { 15 } + 13 q + 16 r\) Given that \(\mathrm { E } \left( X ^ { 3 } \right) = \mathrm { E } \left( X ^ { 2 } \right) + \mathrm { E } ( 6 X )\)
3. find the value of \(q\) and the value of \(r\)
4. Hence find \(\mathrm { P } \left( X ^ { 3 } > X ^ { 2 } + 6 X \right)\)

1. The continuous random variable X has probability density function

$$f ( x ) = \begin{cases} \frac { 1 } { 8 } & 1 \leqslant x \leqslant 9 \\ 0 & \text { otherwise } \end{cases}$$

1. Write down the name given to this distribution. The continuous random variable \(Y = 5 - 2 X\)
2. Find \(\mathrm { P } ( Y > 0 )\)
3. Find \(\mathrm { E } ( Y )\)
4. Find \(\mathrm { P } ( Y < 0 \mid X < 7.5 )\)

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The random variable \(X\) has a continuous uniform distribution over the interval [5,a], where \(a\) is a constant.
Given that \(\operatorname { Var } ( X ) = \frac { 27 } { 4 }\)

1. show that \(a = 14\) The continuous random variable \(Y\) has probability density function $$f ( y ) = \left\{ \begin{array} { c c } \frac { 1 } { 20 } ( 2 y - 3 ) & 2 \leqslant y \leqslant 6 \\ 0 & \text { otherwise } \end{array} \right.$$ The random variable \(T = 3 \left( X ^ { 2 } + X \right) + 2 Y\)
2. Show that \(\mathrm { E } ( T ) = \frac { 9857 } { 30 }\)

1. The continuous random variable \(X\) has cumulative distribution function

$$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x < 4 \\ p x - k \sqrt { x } & 4 \leqslant x \leqslant 9 \\ 1 & x > 9 \end{array} \right.$$ where \(p\) and \(k\) are constants.

1. Find the value of \(p\) and the value of \(k\). Given that \(\mathrm { E } ( X ) = \frac { 119 } { 18 }\)
2. show that \(\operatorname { Var } ( X ) = 2.05\) to 3 significant figures.
3. Write down the mode of \(X\).
4. Find the exact value of the constant \(a\) such that \(\mathrm { P } ( X \leqslant a ) = \frac { 7 } { 27 }\)

1. A random variable \(X\) has probability density function given by

$$f ( x ) = \left\{ \begin{array} { c c } 0.8 - 6.4 x ^ { - 3 } & 2 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{array} \right.$$ The median of \(X\) is \(m\)

1. Show that \(m ^ { 3 } - 3.625 m ^ { 2 } + 4 = 0\)
2. 1. Find \(\mathrm { f } ^ { \prime } ( x )\)
   2. Explain why the mode of \(X\) is 4 Given that \(\mathrm { E } \left( X ^ { 2 } \right) = 10.5\) to 3 significant figures,
3. find \(\operatorname { Var } ( X )\), showing your working clearly.

1. A continuous random variable \(X\) has probability density function

$$f ( x ) = \left\{ \begin{array} { c c } \frac { x } { 16 } \left( 9 - x ^ { 2 } \right) & 1 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{array} \right.$$

1. Find the cumulative distribution function of \(X\)
2. Calculate \(\mathrm { P } ( X > 1.8 )\)
3. Use calculus to find \(\mathrm { E } \left( \frac { 3 } { X } + 2 \right)\)
4. Show that the mode of \(X\) is \(\sqrt { 3 }\)

1. The continuous random variable \(Y\) has probability density function

$$f ( y ) = \left\{ \begin{array} { c c } \frac { 1 } { 24 } ( y + 2 ) ( 4 - y ) & 0 \leqslant y \leqslant 3 \\ 0 & \text { otherwise } \end{array} \right.$$

1. Show that the mode of \(Y\) is 1 , justifying your reasoning. Given that \(\mathrm { P } ( Y < 1 ) = \frac { 13 } { 36 }\)
2. determine whether the median of \(Y\) is less than, equal to, or greater than 2 Give a reason for your answer. Given that \(\mathrm { E } \left( Y ^ { 2 } \right) = \frac { 213 } { 80 }\)
3. find, using algebraic integration, \(\operatorname { Var } ( 2 Y )\)

1. The continuous random variable \(X\) has probability density function

$$f ( x ) = \begin{cases} \frac { 1 } { 18 } ( 11 - 2 x ) & 1 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$

1. Find \(\mathrm { P } ( \mathrm { X } < 3 )\)
2. State, giving a reason, whether the upper quartile of \(X\) is greater than 3, less than 3 or equal to 3 Given that \(\mathrm { E } ( \mathrm { X } ) = \frac { 9 } { 4 }\)
3. use algebraic integration to find \(\operatorname { Var } ( \mathrm { X } )\) The cumulative distribution function of \(X\) is given by $$F ( x ) = \left\{ \begin{array} { l r } 0 & x < 1 \\ \frac { 1 } { 18 } \left( 11 x - x ^ { 2 } + c \right) & 1 \leqslant x \leqslant 4 \\ 1 & x > 4 \end{array} \right.$$
4. Show that \(\mathrm { c } = - 10\)
5. Find the median of \(X\), giving your answer to 3 significant figures. \section*{Q uestion 2 continued}

The continuous random variable \(X\) is uniformly distributed over the interval \([ - 3,5 ]\).

1. Sketch the probability density function \(\mathrm { f } ( \mathrm { x } )\) of X .
2. Find the value of k such that \(\mathrm { P } ( \mathrm { X } < 2 [ \mathrm { k } - \mathrm { X } ] ) = 0.25\)
3. Use algebraic integration to show that \(\mathrm { E } \left( \mathrm { X } ^ { 3 } \right) = 17\)

4 The continuous random variable \(X\) has cumulative distribution function given by $$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x \leqslant 0 \\ k \left( x ^ { 3 } - \frac { 3 } { 8 } x ^ { 4 } \right) & 0 < x \leqslant 2 \\ 1 & x > 2 \end{array} \right.$$ where \(k\) is a constant.

1. Show that \(k = \frac { 1 } { 2 }\)
2. Showing your working clearly, use calculus to find
   1. \(\mathrm { E } ( X )\)
   2. the mode of \(X\)
3. Describe, giving a reason, the skewness of the distribution of \(X\)

5 \begin{figure}[h] \end{figure} The random variable \(X\) has probability density function \(\mathrm { f } ( x )\) and Figure 1 shows a sketch of \(\mathrm { f } ( x )\) where $$f ( x ) = \left\{ \begin{array} { c c } k ( 1 - \cos x ) & 0 \leqslant x \leqslant 2 \pi \\ 0 & \text { otherwise } \end{array} \right.$$

1. Show that \(k = \frac { 1 } { 2 \pi }\) The random variable \(Y \sim \mathrm {~N} \left( \mu , \sigma ^ { 2 } \right)\) and \(\mathrm { E } ( Y ) = \mathrm { E } ( X )\) The probability density function of \(Y\) is \(g ( y )\), where $$g ( y ) = \frac { 1 } { \sigma \sqrt { 2 \pi } } e ^ { - \frac { 1 } { 2 } \left( \frac { y - \mu } { \sigma } \right) ^ { 2 } } \quad - \infty < y < \infty$$ Given that \(\mathrm { g } ( \mu ) = \mathrm { f } ( \mu )\)
2. find the exact value of \(\sigma\)
3. Calculate the error in using \(\mathrm { P } \left( \frac { \pi } { 2 } < Y < \frac { 3 \pi } { 2 } \right)\) as an approximation to \(\mathrm { P } \left( \frac { \pi } { 2 } < X < \frac { 3 \pi } { 2 } \right)\)

8 A circle, centre \(O\), has radius \(x \mathrm {~cm}\), where \(x\) is an observation from the random variable \(X\) which has a rectangular distribution on \([ 0 , \pi ]\)

1. Find the probability that the area of the circle is greater than \(10 \mathrm {~cm} ^ { 2 }\)
2. State, giving a reason, whether the median area of the circle is greater or less than \(10 \mathrm {~cm} ^ { 2 }\) The triangle \(O A B\) is drawn inside the circle with \(O A\) and \(O B\) as radii of length \(x \mathrm {~cm}\) and angle \(A O B x\) radians.
3. Use algebraic integration to find the expected value of the area of triangle \(O A B\). Give your answer as an exact value.

1. The continuous random variable \(X\) has cumulative distribution function given by

$$\mathrm { F } ( x ) = \left\{ \begin{array} { c r } 0 & x < 2 \\ 1.25 - \frac { 2.5 } { x } & 2 \leqslant x \leqslant 10 \\ 1 & x > 10 \end{array} \right.$$

1. Find \(\mathrm { P } ( \{ X < 5 \} \cup \{ X > 8 \} )\)
2. Find the median of \(X\).
3. Find \(\mathrm { E } \left( X ^ { 2 } \right)\)
4. 1. Sketch the probability density function of \(X\).
   2. Describe the skewness of the distribution of \(X\).

1. A rectangle is to have an area of \(40 \mathrm {~cm} ^ { 2 }\)

The length of the rectangle, \(L \mathrm {~cm}\), follows a continuous uniform distribution over the interval [4, 10] Find the expected value of the perimeter of the rectangle.
Use algebraic integration, rather than your calculator, to evaluate any definite integrals.

The random variable \(R\) has a continuous uniform distribution over the interval \([ 2,10 ]\)

1. Write down the probability density function \(\mathrm { f } ( r )\) of \(R\) A sphere of radius \(R \mathrm {~cm}\) is formed.
   The surface area of the sphere, \(S \mathrm {~cm} ^ { 2 }\), is given by \(S = 4 \pi R ^ { 2 }\)
2. Show that \(\mathrm { E } ( S ) = \frac { 496 \pi } { 3 }\) The volume of the sphere, \(V \mathrm {~cm} ^ { 3 }\), is given by \(V = \frac { 4 } { 3 } \pi R ^ { 3 }\)
3. Find, using algebraic integration, the expected value of \(V\)

The random variable \(G\) has a continuous uniform distribution over the interval \([ - 3,15 ]\)

1. Calculate \(\mathrm { P } ( G > 12 )\) The random variable \(H\) has a continuous uniform distribution over the interval [2, w] The random variables \(G\) and \(H\) are independent and \(\mathrm { E } ( H ) = 10\)
2. Show that the probability that \(G\) and \(H\) are both greater than 12 is \(\frac { 1 } { 16 }\) The random variable \(A\) is the area on a coordinate grid bounded by $$\begin{aligned} & y = - 3 \\ & y = - 4 | x | + k \end{aligned}$$ where \(k\) is a value from the continuous uniform distribution over the interval [5,10]
3. Calculate the expected value of \(A\)

The three independent random variables \(A , B\) and \(C\) each have a continuous uniform distribution over the interval \([ 0,5 ]\).

1. Find the probability that \(A , B\) and \(C\) are all greater than 3 The random variable \(Y\) represents the maximum value of \(A , B\) and \(C\).
   The cumulative distribution function of \(Y\) is $$\mathrm { F } ( y ) = \begin{cases} 0 & y < 0 \\ \frac { y ^ { 3 } } { 125 } & 0 \leqslant y \leqslant 5 \\ 1 & y > 5 \end{cases}$$
2. Using algebraic integration, show that \(\operatorname { Var } ( Y ) = 0.9375\)
3. Find the mode of \(Y\), giving a reason for your answer.
4. Describe the skewness of the distribution of \(Y\). Give a reason for your answer.
5. Find the value of \(k\) such that \(\mathrm { P } ( k < Y < 2 k ) = 0.189\)

4 The discrete random variable \(X\) has the distribution \(\mathrm { U } ( n )\).

1. Use the results \(\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )\) and \(\mathrm { E } ( X ) = \frac { n + 1 } { 2 }\) to show that \(\operatorname { Var } ( X ) = \frac { 1 } { 12 } \left( n ^ { 2 } - 1 \right)\). It is given that \(\mathrm { E } ( X ) = 13\).
2. Find the value of \(n\).
3. Find \(\mathrm { P } ( X < 7.5 )\). It is given that \(\mathrm { E } ( a X + b ) = 10\) and \(\operatorname { Var } ( a X + b ) = 117\), where \(a\) and \(b\) are positive.
4. Calculate the value of \(a\) and the value of \(b\).

2 A continuous random variable \(X\) has cumulative distribution function given by $$\mathrm { F } ( x ) = \begin{cases} 1 - \frac { 1 } { x ^ { 3 } } & x \geqslant 1 \\ 0 & \text { otherwise } \end{cases}$$ Find \(E ( \sqrt { } X )\).

8 A continuous random variable \(X\) has probability density function given by the following function, where \(a\) is a constant. \(\mathrm { f } ( x ) = \left\{ \begin{array} { l l } \frac { 2 x } { a ^ { 2 } } & 0 \leqslant x \leqslant a , \\ 0 & \text { otherwise. } \end{array} \right\}\) The expected value of \(X\) is 4 .

1. Show that \(a = 6\). Five independent observations of \(X\) are obtained, and the largest of them is denoted by \(M\).
2. Find the cumulative distribution function of \(M\). \section*{OCR} Oxford Cambridge and RSA

4 The continuous random variable \(X\) has the cumulative distribution function $$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x < - c \\ \frac { x + c } { 4 c } & - c \leqslant x \leqslant 3 c \\ 1 & x > 3 c \end{array} \right.$$ where \(c\) is a positive constant.

1. Determine \(\mathrm { P } \left( - \frac { 3 c } { 4 } < X < \frac { 3 c } { 4 } \right)\).
2. Show that the probability density function, \(\mathrm { f } ( x )\), of \(X\) is $$f ( x ) = \left\{ \begin{array} { c c } \frac { 1 } { 4 c } & - c \leqslant x \leqslant 3 c \\ 0 & \text { otherwise } \end{array} \right.$$
3. Hence, or otherwise, find expressions, in terms of \(c\), for:
   1. \(\mathrm { E } ( X )\);
   2. \(\operatorname { Var } ( X )\).

4 Students are each asked to measure the distance between two points to the nearest tenth of a metre.

1. Given that the rounding error, \(X\) metres, in these measurements has a rectangular distribution, explain why its probability density function is $$f ( x ) = \left\{ \begin{array} { c c } 10 & - 0.05 < x \leqslant 0.05 \\ 0 & \text { otherwise } \end{array} \right.$$
2. Calculate \(\mathrm { P } ( - 0.01 < X < 0.02 )\).
3. Find the mean and the standard deviation of \(X\).

6 The continuous random variable \(X\) has the probability density function given by $$f ( x ) = \left\{ \begin{array} { c c } 3 x ^ { 2 } & 0 < x \leqslant 1 \\ 0 & \text { otherwise } \end{array} \right.$$

1. Determine:
   1. \(\mathrm { E } \left( \frac { 1 } { X } \right)\);
      (3 marks)
   2. \(\operatorname { Var } \left( \frac { 1 } { X } \right)\).
2. Hence, or otherwise, find the mean and the variance of \(\left( \frac { 5 + 2 X } { X } \right)\).