Piecewise PDF with k

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 \(X\).

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 \(X\).

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

1. Sketch the graph of f.
2. Show that \(k = \frac { 4 } { 33 }\). \includegraphics[max width=\textwidth, alt={}, center]{e2a45d19-7d48-4aa5-93f9-6ef90f99d7c4-09_2725_35_99_20}
3. Find the cumulative distribution function of \(X\).
4. Find the median value of \(X\).

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

1. Sketch the graph of f.
2. Show that \(k = \frac { 4 } { 33 }\). \includegraphics[max width=\textwidth, alt={}, center]{8b2a13d7-62f4-45a7-84c5-7d5bc870b8ce-09_2725_35_99_20}
3. Find the cumulative distribution function of \(X\).
4. Find the median value of \(X\).

3 The monthly demand for a product, \(X\) thousand units, is modelled by the random variable \(X\) with probability density function given by $$f ( x ) = \begin{cases} a x & 0 \leqslant x \leqslant 1 \\ a ( x - 2 ) ^ { 2 } & 1 < x \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) is a positive constant. Find

1. the value of \(a\),
2. the probability that the monthly demand is at most 1500 units,
3. the expected monthly demand.

6 The continuous random variable \(X\) has probability density function given by $$\mathrm { f } ( x ) = \left\{ \begin{array} { c l } k \sin x & 0 \leqslant x \leqslant \frac { 1 } { 2 } \pi , \\ k \left( 2 - \frac { 2 x } { \pi } \right) & \frac { 1 } { 2 } \pi \leqslant x \leqslant \pi , \\ 0 & \text { otherwise, } \end{array} \right.$$ where \(k\) is a constant.

1. Show that \(k = \frac { 4 } { 4 + \pi }\).
2. Find \(\mathrm { E } ( X )\), correct to 3 significant figures, showing all necessary working.

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

1. Show that \(k = \frac { 3 } { 10 }\).
2. Find \(\mathrm { E } ( X )\).
3. Find the cumulative distribution function of \(X\).
4. Find the upper quartile of \(X\), correct to 3 significant figures. \section*{END OF QUESTION PAPER}

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

$$f ( x ) = \begin{cases} \frac { 1 } { 16 } x ^ { 2 } & 1 \leqslant x < 3 \\ k ( 4 - x ) & 3 \leqslant x \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$

1. Show that \(k = \frac { 11 } { 12 }\)
2. Sketch \(\mathrm { f } ( x )\) for \(1 \leqslant x \leqslant 4\)
3. Write down the mode of \(X\) Given that \(\mathrm { E } ( X ) = \frac { 25 } { 9 }\)
4. use algebraic integration to find \(\operatorname { Var } ( X )\), giving your answer to 3 significant figures. The cumulative distribution function of \(X\) is given by $$\mathrm { F } ( x ) = \left\{ \begin{array} { l r } 0 & x < 1 \\ \frac { 1 } { 48 } \left( x ^ { 3 } + c \right) & 1 \leqslant x < 3 \\ \frac { 11 } { 12 } \left( 4 x - \frac { 1 } { 2 } x ^ { 2 } + d \right) & 3 \leqslant x \leqslant 4 \\ 1 & x > 4 \end{array} \right.$$
5. 1. Find the exact value of \(C\)
   2. Find the exact value of \(d\)
6. Calculate, to 3 significant figures, the upper quartile of \(X\)
   \includegraphics[max width=\textwidth, alt={}]{a814156d-6945-4601-9cae-d28d8ae0db1e-28_2632_1826_121_121}

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

$$f ( x ) = \begin{cases} c ( x + 3 ) & - 3 \leqslant x < 0 \\ \frac { 5 } { 36 } ( 3 - x ) & 0 \leqslant x \leqslant 3 \\ 0 & \text { otherwise } \end{cases}$$ where \(c\) is a positive constant.

1. Show that \(c = \frac { 1 } { 12 }\)
2. 1. Sketch the probability density function.
   2. Explain why the mode of \(X = 0\)
3. Find the cumulative distribution function of \(X\), for all values of \(x\)
4. Find, to 3 significant figures, the value of \(d\) such that \(\mathrm { P } ( X > d \mid X > 0 ) = \frac { 2 } { 5 }\)

   Leave blank	Q7

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

$$f ( x ) = \begin{cases} \frac { 2 x } { 15 } & 0 \leqslant x \leqslant k \\ \frac { 1 } { 5 } ( 5 - x ) & k < x \leqslant 5 \\ 0 & \text { otherwise } \end{cases}$$

1. Showing your working clearly, find the value of \(k\).
2. Write down the mode of \(X\).
3. Find \(\mathrm { P } \left( \left. X \leqslant \frac { k } { 2 } \right\rvert \, X \leqslant k \right)\)

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

$$f ( x ) = \left\{ \begin{array} { c c } \frac { 1 } { 4 } & - \frac { 1 } { 2 } \leqslant x < \frac { 1 } { 2 } \\ 2 x - \frac { 3 } { 4 } & \frac { 1 } { 2 } \leqslant x \leqslant k \\ 0 & \text { otherwise } \end{array} \right.$$ where \(k\) is a positive constant.

1. Sketch the graph of \(\mathrm { f } ( x )\)
2. By forming and solving an equation in \(k\), show that \(k = 1.25\)
3. Use calculus to find \(\mathrm { E } ( X )\)
4. Calculate the interquartile range of \(X\)

8 The continuous random variable \(X\) has probability density function \(\mathrm { f } ( x )\) It is given that \(\mathrm { f } ( x ) = x ^ { 2 }\) for \(0 \leq x \leq 1\) It is also given that \(\mathrm { f } ( x )\) is a linear function for \(1 < x \leq \frac { 3 } { 2 }\) For all other values of \(x , \mathrm { f } ( x ) = 0\) A sketch of the graph of \(y = \mathrm { f } ( x )\) is shown below. \includegraphics[max width=\textwidth, alt={}, center]{c309e27b-5618-4f94-aecd-a55d8756ef03-12_821_1077_758_543} Show that \(\operatorname { Var } ( X ) = 0.0864\) correct to three significant figures. \includegraphics[max width=\textwidth, alt={}, center]{c309e27b-5618-4f94-aecd-a55d8756ef03-14_2491_1755_173_123} Additional page, if required. number Write the question numbers in the left-hand margin.

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 \(2 m ^ { 3 } + 2 m - 1 = 0\).

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 .

A random variable \(X\) has probability density function \(f\) defined by $$f(x) = \begin{cases} \frac{a}{x^2} - \frac{18}{x^3} & 2 \leqslant x < 3, \\ 0 & \text{otherwise}, \end{cases}$$ where \(a\) is a constant.

1. Show that \(a = \frac{27}{2}\). [3]
2. Show that \(\text{E}(X) = \frac{27}{2} \ln \frac{3}{2} - 3\). [3]

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

1. Show that \(k = \frac{1}{2}\). [2]
2. Find \(\text{P}(X > \frac{1}{2})\). [1]
3. Find the mean of \(X\). [3]
4. Find \(a\) such that \(\text{P}(X < a) = \frac{1}{3}\). [3]

A continuous random variable X has probability density function f(x) where $$f(x) = \begin{cases} k(x^3 + 2x + 1), & -1 \leq x \leq 0, \\ 0, & otherwise \end{cases}$$ where k is a positive integer.

1. Show that k = 3. [4]

Find

1. E(X), [4]
2. the cumulative distribution function F(x), [4]
3. P(−0.3 < X < 0.3). [3]

A random variable \(X\) has probability density function given by $$f(x) = \begin{cases} kx^2 & 0 \leq x \leq 2 \\ k\left(1 - \frac{x}{6}\right) & 2 < x \leq 6 \\ 0 & \text{otherwise} \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac{1}{4}\) [4]
2. Write down the mode of \(X\). [1]
3. Specify fully the cumulative distribution function F(\(x\)). [5]
4. Find the upper quartile of \(X\). [4]