5.03c Calculate mean/variance: by integration

3 The waiting time, \(T\) weeks, for a particular operation at a hospital has probability density function given by $$f ( t ) = \begin{cases} \frac { 1 } { 2500 } \left( 100 t - t ^ { 3 } \right) & 0 \leqslant t \leqslant 10 \\ 0 & \text { otherwise } \end{cases}$$

1. Given that \(\mathrm { E } ( T ) = \frac { 16 } { 3 }\), find \(\operatorname { Var } ( T )\).
2. \(10 \%\) of patients have to wait more than \(n\) weeks for their operation. Find the value of \(n\), giving your answer correct to the nearest integer.

2 \includegraphics[max width=\textwidth, alt={}, center]{323cf83a-e23b-494e-a911-856d8f1c92fd-2_483_791_708_676} The diagram shows the graph of the probability density function, f , of a random variable \(X\).

1. Find the value of the constant \(c\).
2. Find the value of \(a\) such that \(\mathrm { P } ( a < X < 1 ) = 0.1\).
3. Find \(\mathrm { E } ( X )\).

4 A random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} k \left( 4 - x ^ { 2 } \right) & - 2 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac { 3 } { 32 }\).
2. Sketch the graph of \(y = \mathrm { f } ( x )\) and hence write down the value of \(\mathrm { E } ( X )\).
3. Find \(\mathrm { P } ( X < 1 )\).

8 \includegraphics[max width=\textwidth, alt={}, center]{c460afa4-1387-421d-87ac-74a64be99714-4_302_517_276_427} \includegraphics[max width=\textwidth, alt={}, center]{c460afa4-1387-421d-87ac-74a64be99714-4_304_508_274_1215} \includegraphics[max width=\textwidth, alt={}, center]{c460afa4-1387-421d-87ac-74a64be99714-4_305_506_717_431} \includegraphics[max width=\textwidth, alt={}, center]{c460afa4-1387-421d-87ac-74a64be99714-4_302_504_717_1217} The diagrams show the probability density functions of four random variables \(W , X , Y\) and \(Z\). Each of the four variables takes values between - 3 and 3 only, and their standard deviations are \(\sigma _ { W } , \sigma _ { X } , \sigma _ { Y }\) and \(\sigma _ { Z }\) respectively.

1. List \(\sigma _ { W } , \sigma _ { X } , \sigma _ { Y }\) and \(\sigma _ { Z }\) in order of size, starting with the largest.
2. The probability density function of \(X\) is given by $$f ( x ) = \begin{cases} \frac { 1 } { 18 } x ^ { 2 } & - 3 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$
   1. Show that \(\sigma _ { X } = 2.32\) correct to 3 significant figures.
   2. Calculate \(\mathrm { P } \left( X > \sigma _ { X } \right)\).
   3. Write down the value of \(\mathrm { P } \left( X > 2 \sigma _ { X } \right)\).

3 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} k x ( 4 - x ) & 0 \leqslant x < 2 \\ k ( 6 - x ) & 2 \leqslant x \leqslant 6 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac { 3 } { 40 }\).
2. Given that \(\mathrm { E } ( X ) = 2.5\), find \(\operatorname { Var } ( X )\).
3. Find the median value of \(X\).

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 \(Y\).

6 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} \frac { 3 } { 28 } \left( e ^ { \frac { 1 } { 2 } x } + 4 e ^ { - \frac { 1 } { 2 } x } \right) & 0 \leqslant x \leqslant 2 \ln 3 \\ 0 & \text { otherwise } \end{cases}$$

1. Find the cumulative distribution function of \(X\).
   The random variable \(Y\) is defined by \(Y = e ^ { \frac { 1 } { 2 } ( X ) }\).
2. Find the probability density function of \(Y\).
3. Find the 30th percentile of \(Y\).
4. Find \(\mathrm { E } \left( Y ^ { 4 } \right)\).
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

7 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \left\{ \begin{array} { c c } \frac { x } { 4 } \left( 4 - x ^ { 2 } \right) & 0 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{array} \right.$$

1. Find \(\operatorname { Var } ( \sqrt { X } )\).
   The continuous random variable \(Y\) is defined by \(Y = X ^ { 2 }\).
2. Find the probability density function of \(Y\).
3. Find the exact value of the median of \(Y\).
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

6 The continuous random variable \(X\) has cumulative distribution function F given by $$F ( x ) = \begin{cases} 0 & x < 0 \\ \frac { 1 } { 60 } \left( 16 x - x ^ { 2 } \right) & 0 \leqslant x \leqslant 6 \\ 1 & x > 6 \end{cases}$$

1. Find the interquartile range of \(X\).
2. Find \(\mathrm { E } \left( X ^ { 3 } \right)\).
   The random variable \(Y\) is such that \(Y = \sqrt { X }\).
3. Find the probability density function of \(Y\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

4 The continuous random variable \(X\) has cumulative distribution function F given by $$F ( x ) = \begin{cases} 0 & x < 2 \\ \frac { 1 } { 60 } x ^ { 2 } - \frac { 1 } { 15 } & 2 \leqslant x \leqslant 8 \\ 1 & x > 8 \end{cases}$$

1. Find \(\mathrm { P } ( 3 \leqslant X \leqslant 6 )\).
2. Find \(\mathrm { E } ( \sqrt { X } )\).
3. Find \(\operatorname { Var } ( \sqrt { X } )\).
4. The random variable \(Y\) is defined by \(Y = X ^ { 3 }\). Find the probability density function of \(Y\).

2 The continuous random variable \(X\) has cumulative distribution function F given by $$F ( x ) = \left\{ \begin{array} { l c } 0 & x < - 1 \\ \frac { 1 } { 2 } ( 1 + x ) ^ { 2 } & - 1 \leqslant x \leqslant 0 \\ 1 - \frac { 1 } { 2 } ( 1 - x ) ^ { 2 } & 0 < x \leqslant 1 \\ 1 & x > 1 \end{array} \right.$$

1. Find the probability density function of \(X\).
2. Find \(\mathrm { P } \left( - \frac { 1 } { 2 } \leqslant X \leqslant \frac { 1 } { 2 } \right)\).
3. Find \(\mathrm { E } \left( X ^ { 2 } \right)\).
4. Find \(\operatorname { Var } \left( X ^ { 2 } \right)\).

3 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} a + \frac { 1 } { 5 } x & 0 \leqslant x < 1 \\ 2 a - \frac { 1 } { 5 } x & 1 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) is a constant.

1. Find the value of \(a\).
2. Find \(\mathrm { E } \left( X ^ { 2 } \right)\).
3. Find the cumulative distribution function of \(X\).

5 The continuous random variable \(X\) has cumulative distribution function F given by $$F ( x ) = \begin{cases} 0 & x < 0 \\ 1 - \frac { 1 } { 144 } ( 12 - x ) ^ { 2 } & 0 \leqslant x \leqslant 12 \\ 1 & x > 12 \end{cases}$$

1. Find the upper quartile of \(X\).
2. Find \(\operatorname { Var } \left( X ^ { 2 } \right)\).
   The random variable \(Y\) is given by \(Y = \sqrt { X }\).
3. Find the probability density function of \(Y\).

4 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} k & 0 \leqslant x < 1 \\ k x & 1 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac { 2 } { 5 }\).
2. Find the interquartile range of \(X\).
3. Find \(\operatorname { Var } ( X )\).

4 \includegraphics[max width=\textwidth, alt={}, center]{a9f9cf66-0734-4316-99ae-c57090d08135-08_353_1141_255_463} The diagram shows the continuous random variable \(X\) with probability density function f given by $$f ( x ) = \begin{cases} \frac { 1 } { 128 } \left( 4 a x - b x ^ { 3 } \right) & 0 \leqslant x \leqslant 4 \\ c & 4 \leqslant x \leqslant 6 \\ 0 & \text { otherwise } \end{cases}$$ where \(a , b\) and \(c\) are constants.
The upper quartile of \(X\) is equal to 4 .

1. Show that \(c = \frac { 1 } { 8 }\) and find the values of \(a\) and \(b\).
2. Find the exact value of the median of \(X\).
3. Find \(\mathrm { E } ( \sqrt { X } )\), giving your answer correct to 2 decimal places.

6 The continuous random variable \(X\) has the following probability density function: $$f ( x ) = \begin{cases} a + b x & 0 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) and \(b\) are constants.

1. Show that \(2 a + 2 b = 1\).
2. It is given that \(\mathrm { E } ( X ) = \frac { 11 } { 9 }\). Use this information to find a second equation connecting \(a\) and \(b\), and hence find the values of \(a\) and \(b\).
3. Determine whether the median of \(X\) is greater than, less than, or equal to \(\mathrm { E } ( X )\).

7 A continuous random variable \(X _ { 1 }\) has probability density function given by $$f ( x ) = \begin{cases} k x & 0 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac { 1 } { 2 }\).
2. Sketch the graph of \(y = \mathrm { f } ( x )\).
3. Find \(\mathrm { E } \left( X _ { 1 } \right)\) and \(\operatorname { Var } \left( X _ { 1 } \right)\).
4. Sketch the graph of \(y = \mathrm { f } ( x - 1 )\).
5. The continuous random variable \(X _ { 2 }\) has probability density function \(\mathrm { f } ( x - 1 )\) for all \(x\). Write down the values of \(\mathrm { E } \left( X _ { 2 } \right)\) and \(\operatorname { Var } \left( X _ { 2 } \right)\).

7 The continuous random variable \(X\) has the probability density function shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{b69b1fe8-790d-4727-a892-8ab2ade08962-3_364_766_1229_699}

1. Find the value of the constant \(k\).
2. Write down the mean of \(X\), and use integration to find the variance of \(X\).
3. Three observations of \(X\) are made. Find the probability that \(X < 9\) for all three observations.
4. The mean of 32 observations of \(X\) is denoted by \(\bar { X }\). State the approximate distribution of \(\bar { X }\), giving its mean and variance. \section*{[Question 8 is printed overleaf.]}

1 Calculate the variance of the continuous random variable with probability density function given by $$f ( x ) = \begin{cases} \frac { 3 } { 37 } x ^ { 2 } & 3 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$

5 A continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} \frac { 1 } { 2 } \pi \sin ( \pi x ) & 0 \leqslant x \leqslant 1 \\ 0 & \text { otherwise } \end{cases}$$

1. Show that this is a valid probability density function. [4]
2. Sketch the curve \(\boldsymbol { y } = \mathbf { f } ( \boldsymbol { x } )\) and write down the value of \(\mathbf { E } \boldsymbol { ( } \boldsymbol { X } \boldsymbol { ) }\). [3]
3. Find the value \(q\) such that \(\mathrm { P } ( X > q ) = 0.75\). [3]
4. Write down an expression, including an integral, for \(\operatorname { Var } ( X )\). (Do not attempt to evaluate the integral.) [2]
5. A student states that " \(X\) is more likely to occur when \(x\) is close to \(\mathrm { E } ( X )\)." Give an improved version of this statement. [1]

7 The time, in minutes, for which a customer is prepared to wait on a telephone complaints line is modelled by the random variable \(X\). The probability density function of \(X\) is given by $$\mathrm { f } ( x ) = \begin{cases} k x \left( 9 - x ^ { 2 } \right) & 0 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac { 4 } { 81 }\).
2. Find \(\mathrm { E } ( X )\).
3. (a) Show that the value \(y\) which satisfies \(\mathrm { P } ( X < y ) = \frac { 3 } { 5 }\) satisfies $$5 y ^ { 4 } - 90 y ^ { 2 } + 243 = 0 .$$ (b) Using the substitution \(w = y ^ { 2 }\), or otherwise, solve the equation in part (a) to find the value of \(y\).

2 The continuous random variable \(X\) takes values in the interval \(0 \leqslant x \leqslant 3\) only with probability density function f . The graph of \(y = \mathrm { f } ( x )\) consists of the two line segments shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{4a6d94a2-66e1-449a-ac0e-1fbada74bb3b-2_524_1287_950_429}

1. Show that \(a = \frac { 2 } { 3 }\).
2. Find the equations of the two line segments.
3. Hence write down the probability density function of \(X\).
4. Find \(\mathrm { E } ( X )\).

6 The lifetime of a particular machine, in months, can be modelled by the random variable \(T\) with probability density function given by $$\mathrm { f } ( t ) = \begin{cases} \frac { 3 } { t ^ { 4 } } & t \geqslant 1 \\ 0 & \text { otherwise. } \end{cases}$$

1. Obtain the (cumulative) distribution function of \(T\).
2. Show that the probability density function of the random variable \(Y\), where \(Y = T ^ { 3 }\), is given by \(\mathrm { g } ( y ) = \frac { 1 } { y ^ { 2 } }\), for \(y \geqslant 1\).
3. Find \(\mathrm { E } ( \sqrt { Y } )\).

7 The continuous random variable \(T\) has probability density function given by $$f ( t ) = \begin{cases} 4 t ^ { 3 } & 0 < t \leqslant 1 \\ 0 & \text { otherwise } \end{cases}$$

1. Obtain the cumulative distribution function of \(T\).
2. Find the cumulative distribution function of \(H\), where \(H = \frac { 1 } { T ^ { 4 } }\), and hence show that the probability density function of \(H\) is given by \(\mathrm { g } ( h ) = \frac { 1 } { h ^ { 2 } }\) over an interval to be stated.
3. Find \(\mathrm { E } \left( 1 + 2 H ^ { - 1 } \right)\).

6 The continuous random variable \(X\) has (cumulative) distribution function given by $$\mathrm { F } ( x ) = \begin{cases} 0 & x < \frac { 1 } { 2 } \\ \frac { 2 x - 1 } { x + 1 } & \frac { 1 } { 2 } \leqslant x \leqslant 2 \\ 1 & x > 2 . \end{cases}$$

1. Given that \(Y = \frac { 1 } { X }\), find the (cumulative) distribution function of \(Y\), and deduce that \(Y\) and \(X\) have identical distributions.
2. Find \(\mathrm { E } ( X + 1 )\) and deduce the value of \(\mathrm { E } \left( \frac { 1 } { X } \right)\).