5.03b Solve problems: using pdf

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

1. The discrete random variable \(Y\) has the following probability distribution

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

1. Write down the value of \(\mathrm { E } ( Y )\) The cumulative distribution function of \(Y\) is \(\mathrm { F } ( y )\) Given that \(F ( 0 ) = \frac { 19 } { 30 }\)
2. show that the value of \(u\) is \(\frac { 4 } { 15 }\) Given also that \(\operatorname { Var } ( Y ) = 37\)
3. find the value of \(q\) and the value of \(r\) The coordinates of a point \(P\) are \(( 12 , Y )\) The random variable \(D\) represents the length of \(O P\)
4. Find the probability distribution of \(D\)

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. \(\operatorname { Var } ( X )\).

7 The continuous random variable \(X\) has the probability density function given by $$f ( x ) = \left\{ \begin{array} { c c } \frac { 1 } { 16 } x ^ { 3 } & 0 \leqslant x \leqslant 2 \\ \frac { 1 } { 6 } ( 5 - x ) & 2 \leqslant x \leqslant 5 \\ 0 & \text { otherwise } \end{array} \right.$$

1. Sketch the graph of f.
2. Prove that the cumulative distribution function of \(X\) for \(2 \leqslant x \leqslant 5\) can be written in the form $$\mathrm { F } ( x ) = 1 - \frac { 1 } { 12 } ( 5 - x ) ^ { 2 }$$
3. Hence, or otherwise, determine \(\mathrm { P } ( X \geqslant 3 \mid X \leqslant 4 )\).

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 \(\left( \frac { 5 + 2 X } { X } \right)\).

4 The continuous random variable \(X\) has probability density function given by $$f ( x ) = \left\{ \begin{array} { c c } \frac { 1 } { 2 } & 0 \leqslant x \leqslant 1 \\ \frac { 3 - x } { 4 } & 1 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{array} \right.$$

1. Sketch the graph of f.
2. Explain why the value of \(\eta\), the median of \(X\), is 1 .
3. Show that the value of \(\mu\), the mean of \(X\), is \(\frac { 13 } { 12 }\).
4. Find \(\mathrm { P } ( X < 3 \mu - \eta )\).

1 Let \(X\) be a continuous random variable with probability density function given by $$f ( x ) = \begin{cases} \frac { 3 } { 4 } x ( 2 - x ) & 0 \leq x \leq 2 \\ 0 & \text { otherwise } \end{cases}$$ Find \(\mathrm { P } ( X = 1 )\) Circle your answer.
0 \(\frac { 1 } { 2 }\) \(\frac { 3 } { 4 }\) \(\frac { 27 } { 32 }\)

5 The diagram shows a graph of the probability density function of the random variable \(X\). \includegraphics[max width=\textwidth, alt={}, center]{313cd5ce-07ff-4781-a134-565b8b221145-05_574_1086_406_479} 5

1. State the mode of \(X\).
   5
2. Find the probability density function of \(X\).

4

1. \(\text { Find } \mathrm { P } ( X > 1 )\) [0pt] [3 marks]
   4
2. 
   [0pt] [3 marks]
   □
   4
3. Find \(\mathrm { E } \left( 2 X ^ { - 1 } - 3 \right)\)

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

6 The continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} \frac { 3 x } { 44 } + \frac { 1 } { 22 } & 1 \leq x \leq 5 \\ 0 & \text { otherwise } \end{cases}$$ 6

1. Find \(\mathrm { P } ( X > 2 )\) [0pt] [2 marks]
   6
2. Find the upper quartile of \(X\) Give your answer to two decimal places.
   6
3. Find \(\operatorname { Var } \left( 44 X ^ { - 3 } \right)\) Give your answer to three decimal places.

4 A random variable \(X\) has a rectangular distribution. The mean of \(X\) is 3 and the variance of \(X\) is 3
4

1. Determine the probability density function of \(X\).
   Fully justify your answer. 4
2. A 6 metre clothes line is connected between the point \(P\) on one building and the point \(Q\) on a second building. Roy is concerned the clothes line may break. He uses the random variable \(X\) to model the distance in metres from \(P\) where the clothes line breaks. 4 (b) (i) State a criticism of Roy's model. 4 (b) (ii) On the axes below, sketch the probability density function for an alternative model for the clothes line. \includegraphics[max width=\textwidth, alt={}, center]{3219e2fe-7757-469a-9d0d-654b3e180e8d-05_584_1162_1210_438}

9 The continuous random variable \(X\) has the cumulative distribution function shown below. $$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x < 0 \\ \frac { 1 } { 62 } \left( 4 x ^ { 3 } + 6 x ^ { 2 } + 3 x \right) & 0 \leq x \leq 2 \\ 1 & x > 2 \end{array} \right.$$ The discrete random variable \(Y\) has the probability distribution shown below. The random variables \(X\) and \(Y\) are independent.
Find the exact value of \(\mathrm { E } \left( X ^ { 3 } + Y \right)\).

2 The continuous random variable \(X\) takes values in the interval \(- 1 \leq x \leq 1\) and has probability density function $$f ( x ) = \left\{ \begin{array} { l r } a & - 1 \leq x < 0 \\ a + x ^ { 2 } & 0 \leq x \leq 1 \end{array} \right.$$ where \(a\) is a constant.

1. (A) Sketch the probability density function.
   (B) Show that \(a = \frac { 1 } { 3 }\).
2. Find
   (A) \(\mathrm { P } \left( X < \frac { 1 } { 2 } \right)\),
   (B) the mean of \(X\).
3. Show that the median of \(X\) satisfies the equation \(2 m ^ { 3 } + 2 m - 1 = 0\).

3. A string of length 60 cm is cut a random point.

1. Name a distribution, including parameters, that can be used to model the length of the longer piece of string and find its mean and variance.
2. The longer string is shaped to form the perimeter of a circle. Find the probability that the area of the circle is greater than \(100 \mathrm {~cm} ^ { 2 }\).

4 The continuous random variable \(X\) has probability density function $$\mathrm { f } ( x ) = \begin{cases} \frac { k } { x ^ { n } } & x \geqslant 1 \\ 0 & \text { otherwise } \end{cases}$$ where \(n\) and \(k\) are constants and \(n\) is an integer greater than 1 .

1. Find \(k\) in terms of \(n\).
2. 1. When \(n = 4\), find the cumulative distribution function of \(X\).
   2. Hence determine \(\mathrm { P } ( X > 7 \mid X > 5 )\) when \(n = 4\).
3. Determine the values of \(n\) for which \(\operatorname { Var } ( X )\) is not defined.

1. 1. The continuous random variable \(X\) is uniformly distributed over the interval \([ a , b ]\)

Given that

• \(\mathrm { P } ( X > 27 ) = \frac { 3 } { 4 }\)
• \(\operatorname { Var } ( X ) = 300\)
  1. find the value of \(a\) and the value of \(b\)

Given also that $$4 \times \mathrm { P } ( X < k - 10 ) = \mathrm { P } ( X > k + 20 )$$

find the value of \(k\) (ii) A piece of wire of length 42 cm is cut into 2 pieces at a random point. Each of the two pieces of the wire is bent to form the outline of a square.
Find the probability that the side length of the larger square minus the side length of the smaller square will be greater than 2 cm .

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

$$f ( x ) = \begin{cases} \frac { 1 } { 4 } ( 3 - x ) & 1 \leqslant x \leqslant 2 \\ \frac { 1 } { 4 } & 2 < x \leqslant 3 \\ \frac { 1 } { 4 } ( x - 2 ) & 3 < x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$ The cumulative distribution function of \(X\) is \(\mathrm { F } ( x )\)

1. Show that \(\mathrm { F } ( x ) = \frac { 1 } { 4 } \left( 3 x - \frac { x ^ { 2 } } { 2 } \right) - \frac { 5 } { 8 }\) for \(1 \leqslant x \leqslant 2\)
2. Find \(\mathrm { F } ( x )\) for all values of \(x\)
3. Find \(\mathrm { P } ( 1.2 < X < 3.1 )\)

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

$$f ( x ) = \begin{cases} a x & 0 \leqslant x \leqslant 4 \\ b x + c & 4 < x \leqslant 8 \\ 0 & \text { otherwise } \end{cases}$$ where \(a\), \(b\) and \(c\) are constants. \begin{figure}[h] \end{figure} Figure 1 shows the graph of the probability density function \(\mathrm { f } ( x )\) The graph consists of two straight line segments of equal length joined at the point where \(x = 4\)

1. Show that \(a = \frac { 1 } { 16 }\)
2. Hence find
   1. the value of \(b\)
   2. the value of \(c\)
3. Using algebraic integration, show that \(\operatorname { Var } ( X ) = \frac { 8 } { 3 }\)
4. Find, to 2 decimal places, the lower quartile and the upper quartile of \(X\) A statistician claims that $$\mathrm { P } ( - \sigma < X - \mu < \sigma ) > 0.5$$ where \(\mu\) and \(\sigma\) are the mean and standard deviation of \(X\)
5. Show that the statistician's claim is correct.

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 \(\operatorname { Var } ( X )\).

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 $$f ( x ) = \begin{cases} \frac { 4 } { 27 } x ^ { 2 } ( 3 - x ) & 0 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$

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. Given that \(\sigma ^ { 2 } = 0.36\), find \(\mathrm { P } ( | X - \mu | < \sigma )\), where \(\mu\) and \(\sigma ^ { 2 }\) denote the mean and the variance of \(X\) respectively.

6 A rectangle of area \(Y \mathrm {~m} ^ { 2 }\) has a perimeter of 16 m and a side of length \(X \mathrm {~m}\), where \(X\) is a random variable with probability density function, f, given by $$f ( x ) = \begin{cases} \frac { 1 } { 2 } & 0 \leqslant x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$

1. Obtain the cumulative distribution function, F , of \(X\).
2. Show that $$16 - Y = ( 4 - X ) ^ { 2 }$$ and deduce that the probability density function of the random variable \(Y\) is $$g ( y ) = \begin{cases} \frac { 1 } { 4 \sqrt { 16 - y } } & 0 \leqslant y \leqslant 12 \\ 0 & \text { otherwise } \end{cases}$$
3. Find the median of \(Y\).
4. Find \(\mathrm { E } ( Y )\).

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.

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 \(\operatorname { Var } ( Y )\).