Continuous Probability Distributions and Random Variables

1 The continuous random variable \(X\) has probability density function given by $$\mathrm { f } ( x ) = \begin{cases} a & 0 \leqslant x \leqslant 1 ,
\frac { a } { x ^ { 2 } } & x > 1 ,
0 & \text { otherwise. } \end{cases}$$ Find the value of the constant \(a\).

1
\includegraphics[max width=\textwidth, alt={}, center]{879cb813-2380-47a7-bd96-cad0a74d0b4d-2_369_531_255_806} The diagram shows the graph of the probability density function, f , of a random variable \(X\). Find the median of \(X\).

6 The probability density function, f , of a random variable \(X\) is given by $$\mathrm { f } ( x ) = \begin{cases} k \left( 6 x - x ^ { 2 } \right) & 0 \leqslant x \leqslant 6
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.
State the value of \(\mathrm { E } ( X )\) and show that \(\operatorname { Var } ( X ) = \frac { 9 } { 5 }\).

1. The graph shows the probability density function \(\mathrm { f } ( x )\) of the continuous random variable \(X\)
   \includegraphics[max width=\textwidth, alt={}, center]{128c408d-3e08-4f74-8f19-d33ecd5c882f-04_951_1365_322_331}
   1. Find \(\mathrm { P } ( X < 4 )\)
   2. Specify the cumulative distribution function of \(X\) for \(7 \leqslant x \leqslant 11\)

1 Calculate the variance of the continuous random variable with probability density function given by $$f ( x ) = \begin{cases} \frac { 3 } { 37 } x ^ { 2 } & 3 \leqslant x \leqslant 4
0 & \text { otherwise } \end{cases}$$

1
\includegraphics[max width=\textwidth, alt={}, center]{cfffe79d-91c9-48b8-a3e6-887d7891441d-2_478_691_260_724} The random variable \(X\) has probability density function, f , as shown in the diagram, where \(a\) is a constant. Find the value of \(a\) and hence show that \(\mathrm { E } ( X ) = 0.943\) correct to 3 significant figures. [5]

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

6 A continuous random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} 3 ( 1 - x ) ^ { 2 } & 0 \leqslant x \leqslant 1
0 & \text { otherwise } \end{cases}$$ Find

1. \(\mathrm { P } ( X > 0.5 )\),
2. the mean and variance of \(X\).

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

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

1 A random variable \(X\) has probability density function f , where $$f ( x ) = \begin{cases} \frac { 3 } { 2 } \left( 1 - x ^ { 2 } \right) & 0 \leqslant x \leqslant 1
0 & \text { otherwise } \end{cases}$$ Find \(\mathrm { E } ( X )\).

4 The random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { k } { \sqrt { } x } & 0 < x \leqslant a
0 & \text { otherwise } \end{cases}$$ where \(k\) and \(a\) are constants. It is given that \(\mathrm { E } ( X ) = 3\).

1. Find the value of \(a\) and show that \(k = \frac { 1 } { 6 }\).
2. Find the median of \(X\).

6 The probability density function of the continuous random variable \(X\) is given by
\(f ( x ) = \begin{cases} 2 ( 1 + a x ) & 0 \leqslant x \leqslant 1 ,
0 & \text { otherwise } , \end{cases}\)
where \(a\) is a constant.

1. Show that \(a = - 1\).
2. Find the cumulative distribution function of \(X\).
3. Find \(\mathrm { P } ( X < 0.5 )\).
4. Show that \(\mathrm { E } ( X )\) is greater than the median of \(X\).

4
\includegraphics[max width=\textwidth, alt={}, center]{c7cbd61b-9a62-494a-b595-f624ec5c0bea-2_351_561_1562_794} The diagram shows the graph of the probability density function, f , of a random variable \(X\) which takes values between 0 and 2 only.

1. Find \(\mathrm { P } ( 1 < X < 1.5 )\).
2. Find the median of \(X\).
3. Find \(\mathrm { E } ( X )\).

7 A continuous random variable \(X\) has probability density function $$f ( x ) = \left\{ \begin{array} { c c } a x ^ { - 3 } + b x ^ { - 4 } & x \geqslant 1
0 & \text { otherwise } \end{array} \right.$$ where \(a\) and \(b\) are constants.

1. Explain what the letter \(x\) represents. It is given that \(\mathrm { P } ( X > 2 ) = \frac { 3 } { 16 }\).
2. Show that \(a = 1\), and find the value of \(b\).
3. Find \(\mathrm { E } ( X )\).

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

4. The continuous random variable \(X\) has cumulative distribution function given by $$\mathrm { F } ( x ) = \left\{ \begin{array} { c c } 0 & x \leqslant 0
k \left( x ^ { 3 } - \frac { 3 } { 8 } x ^ { 4 } \right) & 0 < x \leqslant 2
1 & x > 2 \end{array} \right.$$ where \(k\) is a constant.

1. Show that \(k = \frac { 1 } { 2 }\)
2. Showing your working clearly, use calculus to find
   1. \(\mathrm { E } ( X )\)
   2. the mode of \(X\)

7 The continuous random variable \(X\) has probability density function
\(f ( x ) = \begin{cases} k x ^ { n } & 0 \leqslant x \leqslant 1 ,
0 & \text { otherwise, } \end{cases}\)
where \(k\) is a constant and \(n\) is a parameter whose value is positive. It is given that the median of \(X\) is 0.8816 correct to 4 decimal places. Ten independent observations of \(X\) are obtained. Find the expected number of observations that are less than 0.8 .

4

1. \(\text { Find } \mathrm { P } ( X > 1 )\)
   [0pt] [3 marks]
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2. 
   [0pt] [3 marks]
   \end{tabular}} & Do not write outside the box
   \hline \end{tabular} \end{center} □
   4
3. Find \(\mathrm { E } \left( 2 X ^ { - 1 } - 3 \right)\)

2 A random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} \frac { 2 } { 3 } x & 1 \leqslant x \leqslant 2
0 & \text { otherwise } \end{cases}$$

1. Find \(\mathrm { E } ( X )\).
2. Find \(\mathrm { P } ( X < \mathrm { E } ( X ) )\).
3. Hence explain whether the mean of \(X\) is less than, equal to or greater than the median of \(X\).

7 The function \(\mathrm { f } ( x )\) is defined by $$f ( x ) = \begin{cases} \frac { 1 } { 4 } x \left( 4 - x ^ { 2 } \right) & 0 \leqslant x \leqslant 2
0 & \text { otherwise } \end{cases}$$

1. Show that \(\mathrm { f } ( x )\) satisfies the conditions for a probability density function.
2. Find the value of \(a\) such that \(\mathrm { P } ( X < a ) = \frac { 15 } { 16 }\).

3. The function \(\mathrm { f } ( x )\) is defined as $$f ( x ) = \begin{cases} \frac { 1 } { 9 } ( x + 5 ) ( 3 - x ) & 1 \leqslant x \leqslant 4
0 & \text { otherwise } \end{cases}$$ Albert believes that \(\mathrm { f } ( x )\) is a valid probability density function.

1. Sketch \(\mathrm { f } ( x )\) and comment on Albert's belief. The continuous random variable \(Y\) has probability density function given by $$g ( y ) = \begin{cases} k y \left( 12 - y ^ { 2 } \right) & 1 \leqslant y \leqslant 3
   0 & \text { otherwise } \end{cases}$$ where \(k\) is a positive constant.
2. Use calculus to find the mode of \(Y\)
3. Use algebraic integration to find the value of \(k\)
4. Find the median of \(Y\) giving your answer to 3 significant figures.
5. Describe the skewness of the distribution of \(Y\) giving a reason for your answer.

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 \(X\).

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

1. Show that \(k = \frac { 1 } { \ln 2 }\).
2. Find \(a\) such that \(\mathrm { P } ( X < a ) = 0.75\).

1. Show that \(b = \frac { a } { a - 1 }\).
2. Given that the median of \(X\) is \(\frac { 3 } { 2 }\), find the values of \(a\) and \(b\).
3. Use your values of \(a\) and \(b\) from part (ii) to find \(\mathrm { E } ( X )\).

6. A random variable \(X\) has a probability density function given by $$\begin{array} { l l } \mathrm { f } ( x ) = \frac { 4 x ^ { 2 } ( 3 - x ) } { 27 } & 0 \leq x \leq 3
\mathrm { f } ( x ) = 0 & \text { otherwise. } \end{array}$$

1. Find the mode of \(X\).
2. Find the mean of \(X\).
3. Specify completely the cumulative distribution function of \(X\).
4. Deduce that the median, \(m\), of \(X\) satisfies the equation \(m ^ { 4 } - 4 m ^ { 3 } + 13 \cdot 5 = 0\), and hence show that \(1.84 < m < 1.85\).
5. What do these results suggest about the skewness of the distribution?

1. Given that \(\mathrm { P } ( X \leqslant 2 ) = \frac { 1 } { 3 }\), show that \(m = \frac { 1 } { 6 }\) and find the values of \(k\) and \(c\).
2. Find the exact numerical value of the interquartile range of \(X\).

6. The continuous random variable \(X\) has (cumulative) distribution function given by $$F ( x ) = \left\{ \begin{array} { c c } 0 & x < 1
1 - \frac { 1 } { x ^ { 4 } } & x \geq 1 \end{array} \right.$$ a. Show that the probability density function of \(Y\), where \(Y = \frac { 1 } { X ^ { 2 } }\), is given by $$g ( y ) = \left\{ \begin{array} { c c } 2 y & 0 < y \leq 1
0 & \text { otherwise } \end{array} \right.$$ b. Find \(\mathrm { E } ( \sqrt [ 3 ] { Y } )\).

5 The lifetime, \(X\) years, of a certain type of battery has probability density function given by $$f ( x ) = \begin{cases} \frac { k } { x ^ { 2 } } & 1 \leqslant x \leqslant a
0 & \text { otherwise } \end{cases}$$ where \(k\) and \(a\) are positive constants.

1. State what the value of \(a\) represents in this context.
2. Show that \(k = \frac { a } { a - 1 }\).
3. Experience has shown that the longest that any battery of this type lasts is 2.5 years. Find the mean lifetime of batteries of this type.

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

$$f ( g ) = \begin{cases} \frac { 1 } { 15 } ( g + 3 ) & - 1 < g \leqslant 2
\frac { 3 } { 20 } & 2 < g \leqslant 4
0 & \text { otherwise } \end{cases}$$

1. Sketch the graph of \(\mathrm { f } ( \mathrm { g } )\)
2. Find \(\mathrm { P } ( ( 1 \leqslant 2 G \leqslant 6 ) \mid G \leqslant 2 )\) The continuous random variable \(H\) is such that \(\mathrm { E } ( H ) = 12\) and \(\operatorname { Var } ( H ) = 2.4\)
3. Find \(\mathrm { E } \left( 2 H ^ { 2 } + 3 G + 3 \right)\) Show your working clearly.
   (Solutions relying on calculator technology are not acceptable.)

6 The continuous random variable \(X\) has probability density function f given by $$f ( x ) = \begin{cases} \frac { 1 } { 80 } \left( 3 \sqrt { } x - \frac { 8 } { \sqrt { } x } \right) & 4 \leqslant x \leqslant 16
0 & \text { otherwise } \end{cases}$$

1. Find the distribution function of \(X\).
   The random variable \(Y\) is defined by \(Y = \sqrt { } X\).
2. Find the probability density function of \(Y\).

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]
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2. Find the upper quartile of \(X\) Give your answer to two decimal places.
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3. Find \(\operatorname { Var } \left( 44 X ^ { - 3 } \right)\) Give your answer to three decimal places.

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

1. Find the value of \(a\).
2. State the value of \(\mathrm { E } ( X )\).
3. Find the interquartile range of \(X\).
   The variable \(Y\) is related to \(X\) by \(Y = 2 ^ { X }\).
4. Find the probability density function of \(Y\).

7 A factory is supplied with grain at the beginning of each week. The weekly demand, \(X\) thousand tonnes, for grain from this factory is a continuous random variable having the probability density function given by $$f ( x ) = \begin{cases} 2 ( 1 - x ) & 0 \leqslant x \leqslant 1
0 & \text { otherwise } \end{cases}$$ Find

1. the mean value of \(X\),
2. the variance of \(X\),
3. the quantity of grain in tonnes that the factory should have in stock at the beginning of a week, in order to be \(98 \%\) certain that the demand in that week will be met.

2

1. 
   \includegraphics[max width=\textwidth, alt={}, center]{b0960fa7-ddbe-47b7-929e-f62f72f9dc93-04_324_574_264_813} The graph of the function f is a straight line segment from \(( 0,0 )\) to \(( 2,1 )\).
   Show that \(f\) could be a probability density function.
2. 
   \includegraphics[max width=\textwidth, alt={}, center]{b0960fa7-ddbe-47b7-929e-f62f72f9dc93-04_364_592_1466_804} The graph of the function g is a semicircle, centre \(( 0,0 )\), entirely above the \(x\)-axis.
   Given that g is a probability density function, find the radius of the semicircle.
3. 
   \includegraphics[max width=\textwidth, alt={}, center]{b0960fa7-ddbe-47b7-929e-f62f72f9dc93-05_369_826_264_689} The time, \(X\) minutes, taken by a large number of students to complete a test has probability density function h , as shown in the diagram.
   1. Without calculation, use the diagram to explain how you can tell that the median time is less than 15 minutes.
      It is now given that $$h ( x ) = \begin{cases} \frac { 40 } { x ^ { 2 } } - \frac { 1 } { 10 } & 10 \leqslant x \leqslant 20
      0 & \text { otherwise. } \end{cases}$$
   2. Find the mean time.