5.03a Continuous random variables: pdf and cdf

1 The lifetime, \(T\) years, of a mortgage may be modelled by the random variable \(T\) with probability density function \(\mathrm { f } ( t )\), where $$\mathrm { f } ( t ) = \begin{cases} k \sin \left( \frac { 3 } { 32 } t \right) & 0 \leqslant t \leqslant 8 \pi \\ 0 & \text { otherwise } \end{cases}$$

1. Show that \(k = \frac { 3 } { 32 } ( 2 - \sqrt { 2 } )\).
2. Sketch the graph of \(\mathrm { f } ( t )\) and state the mode.

4 It is given that \(X\) and \(Y\) are independent random variables with distributions \(\mathrm { N } \left( \mu _ { x } , \sigma _ { x } ^ { 2 } \right)\) and \(\mathrm { N } \left( \mu _ { y } , \sigma _ { y } ^ { 2 } \right)\) respectively, and that \(W\) is a random variable such that \(W = X + Y\).

1. Use moment generating functions to show that the distribution of \(W\) is \(\mathrm { N } \left( \mu _ { x } + \mu _ { y } , \sigma _ { x } ^ { 2 } + \sigma _ { y } ^ { 2 } \right)\).
2. State the distribution of \(X - Y\). The diameters of the central poles of one brand of rotary clothes lines are normally distributed with mean 3.75 cm and variance \(0.000125 \mathrm {~cm} ^ { 2 }\). The diameters of the cylindrical tubes, into which the central poles fit, are normally distributed with mean 3.85 cm and variance \(0.0001 \mathrm {~cm} ^ { 2 }\). Poles and tubes are chosen at random. The 'clearance' between a tube and a pole is the diameter of the tube minus the diameter of the pole.
3. Find the probability that a pole and tube have a clearance between 0.08 cm and 0.13 cm .
4. Given that a pole and tube have a clearance between 0.08 cm and 0.13 cm , find the probability that the clearance is between 0.11 cm and 0.125 cm .

6 The object distance, \(U \mathrm {~cm}\), and the image distance, \(V \mathrm {~cm}\), for a convex lens of focal length 40 cm are related by the lens law $$\frac { 1 } { U } + \frac { 1 } { V } = \frac { 1 } { 40 } .$$ The random variable \(U\) is uniformly distributed over the interval \(80 \leqslant u \leqslant 120\).

1. Show that the probability density function of \(V\) is given by $$f ( v ) = \begin{cases} \frac { 40 } { ( v - 40 ) ^ { 2 } } & 60 \leqslant v \leqslant 80 \\ 0 & \text { otherwise } \end{cases}$$
2. Find
   1. the median value of \(V\),
   2. the expected value of \(V\).

6 A continuous random variable \(X\) has probability density function $$\mathrm { f } ( x ) = \begin{cases} 4 x \mathrm { e } ^ { - 2 x } & x \geqslant 0 \\ 0 & \text { otherwise } . \end{cases}$$

1. Show that the moment generating function \(\mathrm { M } _ { X } ( t )\) of \(X\) is \(\frac { 4 } { ( 2 - t ) ^ { 2 } }\). You may assume that \(x \mathrm { e } ^ { - k x } \rightarrow 0\) as \(x \rightarrow + \infty\).
2. What condition on \(t\) is needed in finding \(\mathrm { M } _ { X } ( t )\) ?
3. \(Y\) is the sum of three independent observations of \(X\). Find the moment generating function of \(Y\), and use your answer to find \(\operatorname { Var } ( Y )\).

7 A continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} \frac { 3 x ^ { 2 } } { k ^ { 3 } } & 0 \leqslant x \leqslant k \\ 0 & \text { otherwise } \end{cases}$$ where \(k\) is a parameter.

1. Find \(\mathrm { E } ( X )\). Hence show that \(\frac { 4 } { 3 } X\) is an unbiased estimator of \(k\). Three independent observations of \(X\) are denoted by \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\), and the largest value of \(X _ { 1 } , X _ { 2 }\) and \(X _ { 3 }\) is denoted by \(M\).
2. Write down an expression for \(\mathrm { P } ( M \leqslant x )\), and hence show that the probability density function of \(M\) is $$f _ { M } ( x ) = \begin{cases} \frac { 9 x ^ { 8 } } { k ^ { 9 } } & 0 \leqslant x \leqslant k \\ 0 & \text { otherwise } . \end{cases}$$
3. Find \(\mathrm { E } ( M )\) and use your answer to construct an unbiased estimator of \(k\) based on \(M\).

1 The random variable \(X\) has probability density function \(\mathrm { f } ( x )\), 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 \(\operatorname { Var } ( X )\).

5 The random variable \(X\) has probability density function \(\mathrm { f } ( x )\), 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 \(\mathrm { E } \left[ \left( X _ { 1 } - X _ { 2 } \right) ^ { 2 } \right]\).

6 The lengths of time, in years, that sales representatives for a certain company keep their company cars may be modelled by the distribution with probability density function \(\mathrm { f } ( x )\), where $$f ( x ) = \left\{ \begin{array} { c c } \frac { 4 } { 27 } x ^ { 2 } ( 3 - x ) & 0 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{array} \right.$$

1. Draw a sketch of this probability density function.
2. Calculate the mean and the mode of \(X\).
3. Comment briefly on the values obtained in part (ii) in relation to the sketch in part (i).
4. Show that the lower quartile \(\mathrm { Q } _ { 1 }\) of \(X\) satisfies the equation \(\mathrm { Q } _ { 1 } { } ^ { 4 } - 4 \mathrm { Q } _ { 1 } { } ^ { 3 } + 6.75 = 0\), and use an appropriate numerical method to find the value of \(\mathrm { Q } _ { 1 }\) correct to 2 decimal places, showing full details of your method.

4 A continuous random variable \(X\) has probability density function $$\mathrm { f } ( x ) = \begin{cases} a & - 1 \leqslant x < 0 \\ a \left( 1 - x ^ { 2 } \right) & 0 \leqslant x \leqslant 1 \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) is a constant.

1. Find the value of \(a\).
2. Find the cumulative distribution function of \(X\).
3. Determine whether the upper quartile is greater than or less than 0.25 .

4 The continuous random variable \(X\) has cumulative distribution function given by $$\mathrm { F } ( x ) = \begin{cases} 0 & x < 0 \\ \frac { 1 } { 8 } x ^ { 3 } & 0 \leqslant x \leqslant 2 \\ 1 & x > 2 \end{cases}$$

1. Find \(\mathrm { E } ( X )\).
2. Find the probability density function of \(Y\), where \(Y = \frac { 1 } { X ^ { 2 } }\).

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 \(\mathrm { E } \left[ \left( X _ { 1 } - X _ { 2 } \right) ^ { 2 } \right]\).

6 The lengths of time, in years, that sales representatives for a certain company keep their company cars may be modelled by the distribution with probability density function \(\mathrm { f } ( x )\), where $$f ( x ) = \left\{ \begin{array} { c c } \frac { 4 } { 27 } x ^ { 2 } ( 3 - x ) & 0 \leqslant x \leqslant 3 , \\ 0 & \text { otherwise } . \end{array} \right.$$

1. Draw a sketch of this probability density function.
2. Calculate the mean and the mode of \(X\).
3. Comment briefly on the values obtained in part (b) in relation to the sketch in part (a).
4. Show that the lower quartile \(\mathrm { Q } _ { 1 }\) of \(X\) satisfies the equation \(\mathrm { Q } _ { 1 } { } ^ { 4 } - 4 \mathrm { Q } _ { 1 } { } ^ { 3 } + 6.75 = 0\), and use an appropriate numerical method to find the value of \(\mathrm { Q } _ { 1 }\) correct to 2 decimal places, showing full details of your method.

5 The random variable \(X\) has probability density function \(\mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = \begin{cases} k e ^ { - k x } & x \geqslant 0 \\ 0 & x < 0 \end{cases}$$ 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 \(\mathrm { E } \left[ \left( X _ { 1 } - X _ { 2 } \right) ^ { 2 } \right]\).

6 The lengths of time, in years, that sales representatives for a certain company keep their company cars may be modelled by the distribution with probability density function \(\mathrm { f } ( x )\), where $$f ( x ) = \left\{ \begin{array} { c c } \frac { 4 } { 27 } x ^ { 2 } ( 3 - x ) & 0 \leqslant x \leqslant 3 \\ 0 & \text { otherwise. } \end{array} \right.$$

1. Draw a sketch of this probability density function.
2. Calculate the mean and the mode of \(X\).
3. Comment briefly on the values obtained in part (b) in relation to the sketch in part (a).
4. Show that the lower quartile \(\mathrm { Q } _ { 1 }\) of \(X\) satisfies the equation \(\mathrm { Q } _ { 1 } { } ^ { 4 } - 4 \mathrm { Q } _ { 1 } { } ^ { 3 } + 6.75 = 0\), and use an appropriate numerical method to find the value of \(\mathrm { Q } _ { 1 }\) correct to 2 decimal places, showing full details of your method.

9 The service time, \(X\) minutes, for each customer in a post office is modelled by the probability density function given by $$\mathrm { f } ( x ) = \begin{cases} 0.2 \mathrm { e } ^ { - 0.2 x } & x > 0 , \\ 0 & \text { otherwise } . \end{cases}$$ Two customers begin to be served independently at the same instant. The larger of the two service times is \(T\) minutes.

1. By considering the probability that both customers have been served in less than \(t\) minutes, show that the cumulative distribution function of \(T\) is given by $$\mathrm { G } ( t ) = \begin{cases} 1 - 2 \mathrm { e } ^ { - 0.2 t } + \mathrm { e } ^ { - 0.4 t } & t > 0 \\ 0 & \text { otherwise } \end{cases}$$
2. Find the probability that both customers are served within 10 minutes.
3. Find the value of the interquartile range of \(T\).

11 The time, \(T\) years, before a particular type of washing machine breaks down may be taken to have probability density function f given by $$\mathrm { f } ( t ) = \begin{cases} a t \mathrm { e } ^ { - b t } & t > 0 \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) and \(b\) are positive constants. It may be assumed that, if \(n\) is a positive integer, $$\int _ { 0 } ^ { \infty } t ^ { n } \mathrm { e } ^ { - b t } \mathrm {~d} t = \frac { n ! } { b ^ { n + 1 } }$$

1. Records show that the mean of \(T\) is 1.5 . Show that \(b = \frac { 4 } { 3 }\) and find the value of \(a\).
2. Find \(\operatorname { Var } ( T )\).
3. Calculate \(\mathrm { P } ( T < 1.5 )\). State, giving a reason, whether this value indicates that the median of \(T\) is smaller than the mean of \(T\) or greater than the mean of \(T\).

The continuous random variable \(X\) is uniformly distributed over the interval \([ 1 , d ]\). a) The 90 th percentile of \(X\) is 19 . Find the value of \(d\).
b) Calculate the mean and standard deviation of \(X\). with mean \(\mu\) and variance \(\sigma ^ { 2 }\). It is found that \(11 \%\) of the large loaves weigh more than 805 g and that \(20 \%\) of the large loaves weigh less than 795 g .
a) Find the values of \(\mu\) and \(\sigma\). The bakery also produces small loaves with masses, in grams, that are normally distributed with mean 400 and standard deviation 9 . Following a change of management at the bakery, a customer suspects that the mean mass of the small loaves has decreased. The customer weighs the next 15 small loaves that he purchases and calculates their mean mass to be 397 g .
b) Perform a hypothesis test at the \(5 \%\) significance level to investigate the customer's suspicion, assuming the standard deviation, in grams, is still 9.
c) State another assumption you have made in part (b). 5 A medical researcher is investigating possible links between diet and a particular disease. She selects a random sample of 22 countries and records the average daily calorie intake per capita from sugar and the percentage of the population who suffer from this disease. \begin{figure}[h] \end{figure} There are 22 data points and the product moment correlation coefficient is \(0 \cdot 893\).
a) Stating your hypotheses clearly, show that these data could be used to suggest that there is a link between the disease and sugar consumption. The medical researcher realises that her data is from the year 2000. She repeats her investigation with a random sample of 13 countries using new data from the year 2020. She produces the following graph. \begin{figure}[h] \end{figure} b) How should the researcher interpret the new data in the light of the data from 2000? \section*{Section B: Differential Equations and Mechanics} A particle \(P\) moves on a horizontal plane, where \(\mathbf { i }\) and \(\mathbf { j }\) are unit vectors in directions east and north respectively. At time \(t\) seconds, the position vector of \(P\) is given by \(\mathbf { r }\) metres, where $$\mathbf { r } = \left( t ^ { 3 } - 7 t ^ { 2 } \right) \mathbf { i } + \left( 2 t ^ { 2 } - 15 t + 11 \right) \mathbf { j }$$ a) i) Find an expression for the velocity vector of \(P\) at time \(t \mathrm {~s}\).
ii) Determine the value of \(t\) when \(P\) is moving north-east and hence write down the velocity of \(P\) at this value of \(t\).
b) Find the acceleration vector of \(P\) when \(t = 7\). smooth supports at points \(X\) and \(Y\), where \(A X = 0.4 \mathrm {~m}\) and \(A Y = 2.4 \mathrm {~m}\). A particle of mass 8 kg is attached to the rod at a point \(C\), where \(B C = 0.2 \mathrm {~m}\). The reaction of the support at \(Y\) is four times the reaction of the support at \(X\). You may not assume that the rod \(A B\) is uniform.
a) i) Find the magnitude of each of the reaction forces exerted on the rod at \(X\) and \(Y\).
ii) Show that the weight of the rod acts at the midpoint of \(A B\).
b) Is it now possible to determine whether the rod is uniform or non-uniform? Give a reason for your answer. A boy kicks a ball from a point \(O\) on horizontal ground towards a vertical wall \(A B\). The initial speed of the ball is \(23 \mathrm {~ms} ^ { - 1 }\) in a direction that is \(18 ^ { \circ }\) above the horizontal. The diagram below shows a window \(C D\) in the wall \(A B\), such that \(B D = 1.1 \mathrm {~m}\) and \(B C = 2 \cdot 2 \mathrm {~m}\). The horizontal distance from \(O\) to \(B\) is 8 m . \includegraphics[max width=\textwidth, alt={}, center]{9c111615-42d5-4804-8eb3-c20fe8d9faee-07_567_1540_605_274} You may assume that the window will break if the ball strikes it with a speed of at least \(21 \mathrm {~ms} ^ { - 1 }\).
a) Show that the ball strikes the window and determine whether or not the window breaks.
b) Give one reason why your answer to part (a) may be unreliable. The diagram below shows a wooden crate of mass 35 kg being pushed on a rough horizontal floor, by a force of magnitude 380 N inclined at an angle of \(30 ^ { \circ }\) below the horizontal. The crate, which may be modelled as a particle, is moving at a constant speed. \includegraphics[max width=\textwidth, alt={}, center]{9c111615-42d5-4804-8eb3-c20fe8d9faee-08_394_665_573_701}
a) The coefficient of friction between the crate and the floor is \(\mu\). Show that $$\mu = \frac { 190 \sqrt { 3 } } { 533 } .$$ Suppose instead that the crate is pulled with the same force of 380 N inclined at an angle of \(30 ^ { \circ }\) above the horizontal, as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{9c111615-42d5-4804-8eb3-c20fe8d9faee-08_392_663_1425_701}
b) Without carrying out any further calculations, explain why the crate will no longer move at a constant speed.

The continuous random variable \(X\) has distribution function given by $$\text{F}(x) = \begin{cases} 0 & x < 0, \\ 1 - e^{-\frac{x}{4}} & x \geqslant 0. \end{cases}$$ For a random value of \(X\), find the probability that \(2\) lies between \(X\) and \(4X\). [3] Find also the expected value of the width of the interval \((X, 4X)\). [4]

Answer only one of the following two alternatives. **EITHER** \includegraphics{figure_11a} A uniform disc, of mass \(4m\) and radius \(a\), and a uniform ring, of mass \(m\) and radius \(2a\), each have centre \(O\). A wheel is made by fixing three uniform rods, \(OA\), \(OB\) and \(OC\), each of mass \(m\) and length \(2a\), to the disc and the ring, as shown in the diagram. Show that the moment of inertia of the wheel about an axis through \(A\), perpendicular to the plane of the wheel, is \(42ma^2\). [5] The axis through \(A\) is horizontal, and the wheel can rotate freely about this axis. The wheel is released from rest with \(O\) above the level of \(A\) and \(AO\) making an angle of \(30°\) with the horizontal. Find the angular speed of the wheel when \(AO\) is horizontal. [3] When \(AO\) is horizontal the disc becomes detached from the wheel. Find the angle that \(AO\) makes with the horizontal when the wheel first comes to instantaneous rest. [6] **OR** The continuous random variable \(T\) has probability density function given by $$f(t) = \begin{cases} 0 & t < 2, \\ \frac{2}{(t-1)^3} & t \geqslant 2. \end{cases}$$

1. Find the distribution function of \(T\), and find also P\((T > 5)\). [3]
2. Consecutive independent observations of \(T\) are made until the first observation that exceeds \(5\) is obtained. The random variable \(N\) is the total number of observations that have been made up to and including the observation exceeding \(5\). Find P\((N > E(N))\). [3]
3. Find the probability density function of \(Y\), where \(Y = \frac{1}{T-1}\). [8]

The number of flaws in a randomly chosen 100 metre length of ribbon is modelled by a Poisson distribution with mean 1.6. The random variable \(X\) metres is the distance between two successive flaws. Show that the distribution function of \(X\) is given by $$\text{F}(x) = \begin{cases} 1 - e^{-0.016x} & x \geq 0, \\ 0 & x < 0, \end{cases}$$ and deduce that \(X\) has a negative exponential distribution, stating its mean. [4] Find

1. the median distance between two successive flaws, [3]
2. the probability that there is a distance of at least 50 metres between two successive flaws. [2]

The number of flaws in a randomly chosen 100 metre length of ribbon is modelled by a Poisson distribution with mean 1.6. The random variable \(X\) metres is the distance between two successive flaws. Show that the distribution function of \(X\) is given by $$\text{F}(x) = \begin{cases} 1 - e^{-0.016x} & x \geqslant 0, \\ 0 & x < 0, \end{cases}$$ and deduce that \(X\) has a negative exponential distribution, stating its mean. [4] Find

1. the median distance between two successive flaws, [3]
2. the probability that there is a distance of at least 50 metres between two successive flaws. [2]

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

1. Find the distribution function of \(X\). [3]

The random variable \(Y\) is defined by \(Y = (X - 1)^3\).

1. Find the probability density function of \(Y\). [4]
2. Find the median value of \(Y\). [3]

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

1. Find the distribution function of \(X\). [3]
2. The random variable \(Y\) is defined by \(Y = (X - 1)^3\). Find the probability density function of \(Y\). [4]
3. Find the median value of \(Y\). [3]

The continuous random variable \(X\) has probability density function f given by $$\text{f}(x) = \begin{cases} 0 & x < 0, \\ ae^{-x \ln 2} & x \geqslant 0, \end{cases}$$ where \(a\) is a positive constant.

1. Find the value of \(a\). [2]
2. State the value of E\((X)\). [1]
3. Find the interquartile range of \(X\). [4]

The variable \(Y\) is related to \(X\) by \(Y = 2^X\).

1. Find the probability density function of \(Y\). [5]

The continuous random variable \(X\) has probability density function f given by $$f(x) = \begin{cases} \frac{3}{4x^2} + \frac{1}{4} & 1 \leqslant x \leqslant 3, \\ 0 & \text{otherwise}. \end{cases}$$

1. Find the distribution function of \(X\). [3]
2. Find the exact value of the interquartile range of \(X\). [5]