5.03d E(g(X)): general expectation formula

6 \includegraphics[max width=\textwidth, alt={}, center]{5dfbc896-528c-40a6-a296-8a0aae90add4-10_451_469_255_799} The diagram shows the graph of the probability density function, f , of a random variable \(X\). The graph is a quarter circle entirely in the first quadrant with centre \(( 0,0 )\) and radius \(a\), where \(a\) is a positive constant. Elsewhere \(\mathrm { f } ( x ) = 0\).

1. Show that \(a = \frac { 2 } { \sqrt { \pi } }\).
2. Show that \(\mathrm { f } ( x ) = \sqrt { \frac { 4 } { \pi } - x ^ { 2 } }\).
3. Show that \(\mathrm { E } ( X ) = \frac { 8 } { 3 \sqrt { \pi ^ { 3 } } }\).

7

1. \includegraphics[max width=\textwidth, alt={}, center]{3f1a0c67-03a4-4b4f-99c0-4336ba7d56b0-3_255_643_264_790} The diagram shows the graph of the probability density function, f , of a random variable \(X\), where $$f ( x ) = \begin{cases} \frac { 2 } { 9 } \left( 3 x - x ^ { 2 } \right) & 0 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$
   1. State the value of \(\mathrm { E } ( X )\) and find \(\operatorname { Var } ( X )\).
   2. State the value of \(\mathrm { P } ( 1.5 \leqslant X \leqslant 4 )\).
   3. Given that \(\mathrm { P } ( 1 \leqslant X \leqslant 2 ) = \frac { 13 } { 27 }\), find \(\mathrm { P } ( X > 2 )\).
2. A random variable, \(W\), has probability density function given by $$\mathrm { g } ( w ) = \begin{cases} a w & 0 \leqslant w \leqslant b \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) and \(b\) are constants. Given that the median of \(W\) is 2 , find \(a\) and \(b\).

9 The continuous random variable \(T\) has cumulative distribution function \(F ( t ) = \begin{cases} 0 & t < 0 , \\ 1 - \mathrm { e } ^ { - 0.25 t } & t \geqslant 0 . \end{cases}\)

1. Find the cumulative distribution function of \(2 T\).
2. Show that, for constant \(k , \mathrm { E } \left( \mathrm { e } ^ { k t } \right) = \frac { 1 } { 1 - 4 k }\). You should state with a reason the range of values of \(k\) for which this result is valid.
3. \(\quad T\) is the time before a certain event occurs. Show that the probability that no event occurs between time \(T = 0\) and time \(T = \theta\) is the same as the probability that the value of a random variable with the distribution \(\operatorname { Po } ( \lambda )\) is 0 , for a certain value of \(\lambda\). You should state this value of \(\lambda\) in terms of \(\theta\). \section*{END OF QUESTION PAPER}

8 A random sample of 100 students were given a task and the time taken by each student to complete the task was recorded. The maximum time allowed to complete the task was one minute and all students completed the task within the maximum time. The times, \(T\) minutes, for the random sample of students are summarised as follows. \(n = 100 \quad \sum t = 61.88\) A researcher proposes that \(T\) can be modelled by the continuous random variable with probability density function \(f ( t ) = \begin{cases} \alpha t ^ { \alpha - 1 } & 0 \leqslant t \leqslant 1 , \\ 0 & \text { otherwise, } \end{cases}\) where \(\alpha\) is a positive constant. \section*{(a) In this question you must show detailed reasoning.} By finding \(\mathbf { E } ( T )\) according to the researcher's model, determine an approximation for the value of \(\alpha\). Give your answer correct to \(\mathbf { 3 }\) significant figures. Further information about the times taken for the sample of 100 students to complete the task is given in the table. (b) Using the value of \(\alpha\) found in part (a), determine the extent to which the proposed model is a good model. (Do not carry out a goodness of fit test.)

4 The continuous random variable \(X\) has a probability density function given by $$\mathrm { f } ( x ) = \begin{cases} \frac { 1 } { 2 } k ( x - 1 ) & 1 \leqslant x \leqslant 3 \\ k & 3 < x \leqslant 6 \\ \frac { 1 } { 4 } k ( 10 - x ) & 6 < x \leqslant 10 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a positive constant.

1. Sketch \(\mathrm { f } ( x )\) for all values of \(x\)
2. Show that \(k = \frac { 1 } { 6 }\)
3. Specify fully the cumulative distribution function \(\mathrm { F } ( x )\) of \(X\) Given that \(\mathrm { E } ( X ) = \frac { 61 } { 12 }\)
4. find \(\mathrm { P } ( X > \mathrm { E } ( X ) )\)
5. Describe the skewness of the distribution, giving a reason for your answer.
   

5. The continuous random variable \(X\) is uniformly distributed over the interval \(\alpha < x < \beta\).

1. Write down the probability density function of \(X\), for all \(x\).
2. Given that \(\mathrm { E } ( X ) = 2\) and \(\mathrm { P } ( X < 3 ) = \frac { 5 } { 8 }\) find the value of \(\alpha\) and the value of \(\beta\). A gardener has wire cutters and a piece of wire 150 cm long which has a ring attached at one end. The gardener cuts the wire, at a randomly chosen point, into 2 pieces. The length, in cm, of the piece of wire with the ring on it is represented by the random variable \(X\). Find
3. \(\mathrm { E } ( X )\),
4. the standard deviation of \(X\),
5. the probability that the shorter piece of wire is at most 30 cm long.

7. The continuous random variable \(X\) has cumulative distribution function $$\mathrm { F } ( x ) = \begin{cases} 0 , & x < 0 \\ 2 x ^ { 2 } - x ^ { 3 } , & 0 \leqslant x \leqslant 1 \\ 1 , & x > 1 \end{cases}$$

1. Find \(\mathrm { P } ( X > 0.3 )\).
2. Verify that the median value of \(X\) lies between \(x = 0.59\) and \(x = 0.60\).
3. Find the probability density function \(\mathrm { f } ( x )\).
4. Evaluate \(\mathrm { E } ( X )\).
5. Find the mode of \(X\).
6. Comment on the skewness of \(X\). Justify your answer.

1. The continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) given by

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

1. Sketch \(\mathrm { f } ( x )\) for all values of \(x\).
2. Write down the mode of \(X\). Find
3. \(\mathrm { E } ( X )\),
4. the median of \(X\).
5. Comment on the skewness of this distribution. Give a reason for your answer.

2. A continuous random variable \(X\) has cumulative distribution function $$\mathrm { F } ( x ) = \begin{cases} 0 , & x < - 2 \\ \frac { x + 2 } { 6 } , & - 2 \leqslant x \leqslant 4 \\ 1 , & x > 4 \end{cases}$$

1. Find \(\mathrm { P } ( X < 0 )\).
2. Find the probability density function \(\mathrm { f } ( x )\) of \(X\).
3. Write down the name of the distribution of \(X\).
4. Find the mean and the variance of \(X\).
5. Write down the value of \(\mathrm { P } ( X = 1 )\).

1. The continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) given by

$$f ( x ) = \begin{cases} k \left( x ^ { 2 } - 2 x + 2 \right) & 0 < x \leqslant 3 \\ 3 k & 3 < x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac { 1 } { 9 }\)
2. Find the cumulative distribution function \(\mathrm { F } ( x )\).
3. Find the mean of \(X\).
4. Show that the median of \(X\) lies between \(x = 2.6\) and \(x = 2.7\)

4

1. A random variable \(X\) has probability density function defined by $$\mathrm { f } ( x ) = \begin{cases} k & a < x < b \\ 0 & \text { otherwise } \end{cases}$$
   1. Show that \(k = \frac { 1 } { b - a }\).
   2. Prove, using integration, that \(\mathrm { E } ( X ) = \frac { 1 } { 2 } ( a + b )\).
2. The error, \(X\) grams, made when a shopkeeper weighs out loose sweets can be modelled by a rectangular distribution with the following probability density function: $$f ( x ) = \begin{cases} k & - 2 < x < 4 \\ 0 & \text { otherwise } \end{cases}$$
   1. Write down the value of the mean, \(\mu\), of \(X\).
   2. Evaluate the standard deviation, \(\sigma\), of \(X\).
   3. Hence find \(\mathrm { P } \left( X < \frac { 2 - \mu } { \sigma } \right)\).

5 A discrete random variable \(X\) has the probability distribution $$\mathrm { P } ( X = x ) = \left\{ \begin{array} { c l } \frac { x } { 20 } & x = 1,2,3,4,5 \\ \frac { x } { 24 } & x = 6 \\ 0 & \text { otherwise } \end{array} \right.$$

1. Calculate \(\mathrm { P } ( X \geqslant 5 )\).
2. 1. Show that \(\mathrm { E } \left( \frac { 1 } { X } \right) = \frac { 7 } { 24 }\).
   2. Hence, or otherwise, show that \(\operatorname { Var } \left( \frac { 1 } { X } \right) = 0.036\), correct to three decimal places.
3. Calculate the mean and the variance of \(A\), the area of rectangles having sides of length \(X + 3\) and \(\frac { 1 } { X }\).

8 The continuous random variable \(X\) has probability density function given by $$\mathrm { f } ( x ) = \left\{ \begin{array} { c c } \frac { 1 } { 2 } \left( x ^ { 2 } + 1 \right) & 0 \leqslant x \leqslant 1 \\ ( x - 2 ) ^ { 2 } & 1 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{array} \right.$$

1. Sketch the graph of f.
2. Calculate \(\mathrm { P } ( X \leqslant 1 )\).
3. Show that \(\mathrm { E } \left( X ^ { 2 } \right) = \frac { 4 } { 5 }\).
4. 1. Given that \(\mathrm { E } ( X ) = \frac { 19 } { 24 }\) and that \(\operatorname { Var } ( X ) = \frac { 499 } { k }\), find the numerical value of \(k\).
   2. Find \(\mathrm { E } \left( 5 X ^ { 2 } + 24 X - 3 \right)\).
   3. Find \(\operatorname { Var } ( 12 X - 5 )\).

5 The continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} x ^ { 3 } & 0 < x \leq 1 \\ \frac { 9 } { 1696 } x ^ { 3 } \left( x ^ { 2 } + 1 \right) & 1 < x \leq 3 \\ 0 & \text { otherwise } \end{cases}$$ 5

1. Find \(\mathrm { P } ( X < 1.8 )\), giving your answer to three decimal places.
   [0pt] [3 marks]
   5
2. Find the lower quartile of \(X\)

   5 (d)

   5 Show that \(\mathrm { E } \left( \frac { 1 } { X ^ { 2 } } \right) = \frac { 133 } { 212 }\)

3 The moment generating function of a random variable \(X\) is \(( 1 - 2 t ) ^ { - 3 }\).

1. Find the mean and variance of \(X\).
2. \(X _ { 1 }\) and \(X _ { 2 }\) are two independent observations of \(X\). Find \(\mathrm { E } \left[ \left( X _ { 1 } + X _ { 2 } \right) ^ { 3 } \right]\).

5 The random variable \(X\) has probability density function \(\mathrm { f } ( \mathrm { x } )\), where $$\mathrm { f } ( x ) = \begin{cases} k \mathrm { e } ^ { - k x } & x \geqslant 0 \\ 0 & x < 0 \end{cases}$$ and \(k\) is a positive constant.

1. Show that the moment generating function of \(X\) is \(\mathrm { M } _ { X } ( t ) = k ( k - t ) ^ { - 1 } , t < k\).
2. Use the moment generating function to find \(\mathrm { E } ( X )\) and \(\operatorname { Var } ( X )\).
3. Show that the moment generating function of \(- X\) is \(k ( k + t ) ^ { - 1 }\).
4. \(X _ { 1 }\) and \(X _ { 2 }\) are two independent observations of \(X\). Use the moment generating function of \(X _ { 1 } - X _ { 2 }\) to find the value of \(\mathrm { E } \left[ \left( X _ { 1 } - X _ { 2 } \right) ^ { 2 } \right]\).

The continuous random variable \(X\) has probability density function f given by $$f(x) = \begin{cases} \frac{1}{2} & 1 \leq x \leq 3, \\ 0 & \text{otherwise.} \end{cases}$$ The random variable \(Y\) is defined by \(Y = X^3\). Find the distribution function of \(Y\). [5] Sketch the graph of the probability density function of \(Y\). [3] Find the probability that \(Y\) lies between its median value and its mean value. [4]

A continuous random variable \(X\) takes values from 0 to 6 only and has a probability distribution that is symmetrical. Two values, \(a\) and \(b\), of \(X\) are such that P\((a < X < b) = p\) and P\((b < X < 3) = \frac{13}{10}p\), where \(p\) is a positive constant.

1. Show that \(p \leq \frac{5}{23}\). [1]
2. Find P\((b < X < 6 - a)\) in terms of \(p\). [2]

It is now given that the probability density function of \(X\) is \(f\), where $$f(x) = \begin{cases} \frac{1}{36}(6x - x^2) & 0 \leq x \leq 6, \\ 0 & \text{otherwise}. \end{cases}$$

1. Given that \(b = 2\) and \(p = \frac{5}{81}\), find the value of \(a\). [5]

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]

The continuous random variable \(Y\) has probability density function given by $$\text{f}(y) = \begin{cases} -\frac{1}{4}y & -2 < y < 0, \\ \frac{1}{4}y & 0 \leqslant y \leqslant 2, \\ 0 & \text{otherwise.} \end{cases}$$ Find

1. the interquartile range of \(Y\), [4]
2. Var\((Y)\), [5]
3. E\((|Y|)\). [4]

A railway company is investigating operations at a junction where delays often occur. Delays (in minutes) are modelled by the random variable \(T\) with the following cumulative distribution function. $$F(t) = \begin{cases} 0 & t \leq 0 \\ 1 - e^{-\frac{1}{t}} & t > 0 \end{cases}$$

1. Find the median delay and the 90th percentile delay. [5]
2. Derive the probability density function of \(T\). Hence use calculus to find the mean delay. [5]
3. Find the probability that a delay lasts longer than the mean delay. [2]

You are given that the variance of \(T\) is 9.

1. Let \(\overline{T}\) denote the mean of a random sample of 30 delays. Write down an approximation to the distribution of \(\overline{T}\). [3]
2. A random sample of 30 delays is found to have mean 4.2 minutes. Does this cast any doubt on the modelling? [3]

A continuous random variable \(X\) has probability density function \(f\) given by $$f(x) = \begin{cases} \frac{x^2}{a} + b, & 0 \leq x \leq 4 \\ 0 & \text{otherwise} \end{cases}$$ where \(a\) and \(b\) are positive constants. It is given that \(P(X \geq 2) = 0.75\).

1. Show that \(a = 32\) and \(b = \frac{1}{12}\). [5]
2. Find \(E(X)\). [3]
3. Find \(P(X > E(X)|X > 2)\) [4]

The continuous random variable \(X\) has cumulative distribution function given by $$F(x) = \begin{cases} 0 & x < 0, \\ \frac{1}{16}x^2 & 0 \leq x \leq 4, \\ 1 & x > 4. \end{cases}$$

1. The random variable \(Y\) is defined by \(Y = \frac{1}{X^2}\). Find the cumulative distribution function of \(Y\). [5]
2. Show that E\((Y)\) is not defined. [4]