5.03f Relate pdf-cdf: medians and percentiles

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.

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 $$f ( x ) = \begin{cases} k x ^ { 3 } & 0 \leqslant x < 1 , \\ k ( 5 - x ) & 1 \leqslant x \leqslant 5 , \\ 0 & \text { otherwise } , \end{cases}$$ 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 \(X\).

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 \(\mathrm { E } ( X )\).

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 \(y\).

5 The continuous random variable \(X\) has cumulative distribution function given by $$F ( x ) = \begin{cases} 0 & x < 1 , \\ \frac { 1 } { 8 } ( x - 1 ) ^ { 2 } & 1 \leqslant x < 3 , \\ a ( x - 2 ) & 3 \leqslant x < 4 , \\ 1 & x \geqslant 4 , \end{cases}$$ where \(a\) is a positive constant.

1. Find the value of \(a\).
2. Verify that \(C _ { X } ( 8 )\), the 8th percentile of \(X\), is 1.8 .
3. Find the cumulative distribution function of \(Y\), where \(Y = \sqrt { X - 1 }\).
4. Find \(C _ { Y } ( 8 )\) and verify that \(C _ { Y } ( 8 ) = \sqrt { C _ { X } ( 8 ) - 1 }\).

7 The continuous random variable \(X\) has (cumulative) distribution function given by $$\mathrm { F } ( x ) = \begin{cases} 0 & x < 1 \\ 1 - \frac { 1 } { x ^ { 4 } } & x \geqslant 1 \end{cases}$$

1. Find the (cumulative) distribution function, \(\mathrm { G } ( y )\), of the random variable \(Y\), where \(Y = \frac { 1 } { X ^ { 2 } }\).
2. Hence show that the probability density function of \(Y\) is given by $$g ( y ) = \begin{cases} 2 y & 0 < y \leqslant 1 \\ 0 & \text { otherwise } \end{cases}$$
3. Find \(\mathrm { E } ( \sqrt [ 3 ] { Y } )\).

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

1. Find the median value of \(X\).
2. Find the (cumulative) distribution function of \(Y\), where \(Y = \frac { 1 } { X ^ { 2 } }\), and hence find the probability density function of \(Y\).
3. Evaluate \(\mathrm { E } \left( 2 - \frac { 2 } { X ^ { 2 } } \right)\).

5 The continuous random variable \(X\) has a triangular distribution with probability density function given by $$f ( x ) = \left\{ \begin{array} { l r } 1 + x & - 1 \leqslant x \leqslant 0 \\ 1 - x & 0 \leqslant x \leqslant 1 \\ 0 & \text { otherwise } \end{array} \right.$$

1. Show that, for \(0 \leqslant a \leqslant 1\), $$\mathrm { P } ( | X | \leqslant a ) = 2 a - a ^ { 2 } .$$ The random variable \(Y\) is given by \(Y = X ^ { 2 }\).
2. Express \(\mathrm { P } ( Y \leqslant y )\) in terms of \(y\), for \(0 \leqslant y \leqslant 1\), and hence show that the probability density function of \(Y\) is given by $$g ( y ) = \frac { 1 } { \sqrt { } y } - 1 , \quad \text { for } 0 < y \leqslant 1 .$$
3. Use the probability density function of \(Y\) to find \(\mathrm { E } ( Y )\), and show how the value of \(\mathrm { E } ( Y )\) may also be obtained directly using the probability density function of \(X\).
4. Find \(\mathrm { E } ( \sqrt { } Y )\).

1 The continuous random variable \(X\) has probability density function $$f ( x ) = k ( 1 - x ) \quad \text { for } 0 \leqslant x \leqslant 1$$ where \(k\) is a constant.

1. Show that \(k = 2\). Sketch the graph of the probability density function.
2. Find \(\mathrm { E } ( X )\) and show that \(\operatorname { Var } ( X ) = \frac { 1 } { 18 }\).
3. Derive the cumulative distribution function of \(X\). Hence find the probability that \(X\) is greater than the mean.
4. Verify that the median of \(X\) is \(1 - \frac { 1 } { \sqrt { 2 } }\).
5. \(\bar { X }\) is the mean of a random sample of 100 observations of \(X\). Write down the approximate distribution of \(\bar { X }\).

9 A rectangle of area \(A \mathrm {~m} ^ { 2 }\) has a perimeter of 20 m and each of the two shorter sides are of length \(X \mathrm {~m}\), where \(X\) is uniformly distributed between 0 and 2 .

1. Write down an expression for \(A\) in terms of \(X\), and hence show that \(A = 25 - ( X - 5 ) ^ { 2 }\).
2. Write down the probability density function of \(X\).
3. Show that the cumulative distribution function of \(A\) is $$\mathrm { F } ( a ) = \left\{ \begin{array} { l r } 0 & a < 0 , \\ \frac { 1 } { 2 } ( 5 - \sqrt { 25 - a } ) & 0 \leqslant a \leqslant 16 , \\ 1 & a > 16 . \end{array} \right.$$
4. Find the probability density function of \(A\). \section*{END OF QUESTION PAPER} \section*{OCR}

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}

2 The continuous random variable \(U\) has (cumulative) distribution function given by $$\mathrm { F } ( u ) = \begin{cases} \frac { 1 } { 5 } \mathrm { e } ^ { u } & u < 0 \\ 1 - \frac { 4 } { 5 } \mathrm { e } ^ { - \frac { 1 } { 4 } u } & u \geqslant 0 \end{cases}$$

1. Find the upper quartile of \(U\).
2. Find the probability density function of \(U\).

6 The function \(\mathrm { F } ( t )\) is defined as follows. $$\mathrm { F } ( t ) = \begin{cases} 0 & t < 0 \\ \sin ^ { 4 } t & 0 \leqslant t \leqslant \frac { 1 } { 2 } \pi \\ 1 & t > \frac { 1 } { 2 } \pi \end{cases}$$

1. Verify that F is a (cumulative) distribution function. The continuous random variable \(T\) has (cumulative) distribution function \(\mathrm { F } ( t )\).
2. Find the lower quartile of \(T\).
3. Find the (cumulative) distribution function of \(Y\), where \(Y = \sin T\), and obtain the probability density function of \(Y\).
4. Find the expected value of \(\frac { 1 } { Y ^ { 3 } + 2 Y ^ { 4 } }\).

8 The continuous random variable \(S\) has probability density function given by $$f ( s ) = \begin{cases} \frac { 8 } { 3 s ^ { 3 } } & 1 \leqslant s \leqslant 2 \\ 0 & \text { otherwise } \end{cases}$$ An isosceles triangle has equal sides of length \(S\), and the angle between them is \(30 ^ { \circ }\) (see diagram).

1. Find the (cumulative) distribution function of the area \(X\) of the triangle, and hence show that the probability density function of \(X\) is \(\frac { 1 } { 3 x ^ { 2 } }\) over an interval to be stated.
2. Find the median value of \(X\). www.ocr.org.uk after the live examination series.
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6 \includegraphics[max width=\textwidth, alt={}, center]{054e0081-afce-4a87-93f5-650dad40b313-3_508_611_262_719} The diagram shows the probability density function f of the continuous random variable \(T\), given by $$f ( t ) = \begin{cases} a t & 0 \leqslant t \leqslant 1 \\ a & 1 < t \leqslant 4 \\ 0 & \text { otherwise } \end{cases}$$ where \(a\) is a constant.

1. Find the value of \(a\).
2. Obtain the cumulative distribution function of \(T\).
3. Find the cumulative distribution of \(Y\), where \(Y = T ^ { \frac { 1 } { 2 } }\), and hence find the probability density function of \(Y\).

5 The continuous random variable \(Y\) has probability density function given by $$\mathrm { f } ( y ) = \begin{cases} \ln ( y ) & 1 \leqslant y \leqslant \mathrm { e } \\ 0 & \text { otherwise } \end{cases}$$

1. Verify that this is a valid probability density function.
2. Show that the (cumulative) distribution function of \(Y\) is given by $$\mathrm { F } ( y ) = \begin{cases} 0 & y < 1 \\ y \ln y - y + 1 & 1 \leqslant y \leqslant \mathrm { e } \\ 1 & \text { otherwise } \end{cases}$$
3. Verify that the upper quartile of \(Y\) lies in the interval [2.45, 2.46].
4. Find the (cumulative) distribution function of \(X\) where \(X = \ln Y\).

3 The time, in hours, until an electronic component fails is represented by the random variable \(X\). In this question two models for \(X\) are proposed.

1. In one model, \(X\) has cumulative distribution function $$\mathrm { G } ( x ) = \begin{cases} 0 & x \leqslant 0 \\ 1 - \left( 1 + \frac { x } { 200 } \right) ^ { - 2 } & x > 0 \end{cases}$$ (A) Sketch \(\mathrm { G } ( x )\).
   (B) Find the interquartile range for this model. Hence show that a lifetime of more than 454 hours (to the nearest hour) would be classed as an outlier.
2. In the alternative model, \(X\) has probability density function $$\mathrm { f } ( x ) = \begin{cases} \frac { 1 } { 200 } \mathrm { e } ^ { - \frac { 1 } { 200 } x } & x > 0 \\ 0 & \text { elsewhere. } \end{cases}$$ (A) For this model show that the cumulative distribution function of \(X\) is $$\mathrm { F } ( x ) = \begin{cases} 0 & x \leqslant 0 \\ 1 - \mathrm { e } ^ { - \frac { 1 } { 200 } x } & x > 0 \end{cases}$$ (B) Show that \(\mathrm { P } ( X > 50 ) = \mathrm { e } ^ { - 0.25 }\).
   (C) It is observed that a particular component is still working after 400 hours. Find the conditional probability that it will still be working after a further 50 hours (i.e. after a total of 450 hours) given that it is still working after 400 hours.

The continuous random variable \(X\) takes values in the interval \(0 \leqslant x \leqslant 3\) only. For \(0 \leqslant x \leqslant 3\) the graph of its probability density function f consists of two straight line segments meeting at the point \(( 1 , k )\), as shown in the diagram. Find \(k\) and hence show that the distribution function F is given by $$\mathrm { F } ( x ) = \begin{cases} 0 & x \leqslant 0 , \\ \frac { 1 } { 3 } x ^ { 2 } & 0 < x \leqslant 1 , \\ x - \frac { 1 } { 2 } - \frac { 1 } { 6 } x ^ { 2 } & 1 < x \leqslant 3 , \\ 1 & x > 3 . \end{cases}$$ The random variable \(Y\) is given by \(Y = X ^ { 2 }\). Find

1. the probability density function of \(Y\),
2. the median value of \(Y\).

The continuous random variable \(X\) takes values in the interval \(0 \leqslant x \leqslant 3\) only. For \(0 \leqslant x \leqslant 3\) the graph of its probability density function f consists of two straight line segments meeting at the point \(( 1 , k )\), as shown in the diagram. Find \(k\) and hence show that the distribution function F is given by $$\mathrm { F } ( x ) = \begin{cases} 0 & x \leqslant 0 , \\ \frac { 1 } { 3 } x ^ { 2 } & 0 < x \leqslant 1 , \\ x - \frac { 1 } { 2 } - \frac { 1 } { 6 } x ^ { 2 } & 1 < x \leqslant 3 , \\ 1 & x > 3 . \end{cases}$$ The random variable \(Y\) is given by \(Y = X ^ { 2 }\). Find

1. the probability density function of \(Y\),
2. the median value of \(Y\).

9 Cotton cloth is sold from long rolls of cloth. The number of flaws on a randomly chosen piece of cloth of length \(a\) metres has a Poisson distribution with mean \(0.8 a\). The random variable \(X\) is the length of cloth, in metres, between two successive flaws.

1. Explain why, for \(x \geqslant 0 , \mathrm { P } ( X > x ) = \mathrm { e } ^ { - 0.8 x }\).
2. Find the probability that there is at least one flaw in a 4 metre length of cloth.
3. Find
   1. the distribution function of \(X\),
   2. the probability density function of \(X\),
   3. the interquartile range of \(X\).

9 The continuous random variable \(X\) has probability density function given by $$\mathrm { f } ( x ) = \begin{cases} 0 & x < 2 \\ a \mathrm { e } ^ { - ( x - 2 ) } & x \geqslant 2 \end{cases}$$ where \(a\) is a constant. Show that \(a = 1\). Find the distribution function of \(X\) and hence find the median value of \(X\). The random variable \(Y\) is defined by \(Y = \mathrm { e } ^ { X }\). Find

1. the probability density function of \(Y\),
2. \(\mathrm { P } ( Y > 10 )\).

8 The random variable \(X\) has probability density function f given by $$\mathrm { f } ( x ) = \begin{cases} 2 \mathrm { e } ^ { - 2 x } & x \geqslant 0 \\ 0 & \text { otherwise } \end{cases}$$

1. Find the distribution function of \(X\).
2. Find the median value of \(X\). The random variable \(Y\) is defined by \(Y = \mathrm { e } ^ { X }\).
3. Find the probability density function of \(Y\).