5.03c Calculate mean/variance: by integration

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 \(\mathrm { P } ( X < 3 \mu - \eta )\).

8 A continuous random variable \(X\) has probability density function given by $$f ( x ) = \begin{cases} k x ^ { n } & 0 \leqslant x \leqslant 1 \\ 0 & \text { otherwise } \end{cases}$$ where \(n\) and \(k\) are positive constants.

1. Find \(k\) in terms of \(n\).
2. Show that \(\mathrm { E } ( X ) = \frac { n + 1 } { n + 2 }\). It is given that \(n = 3\).
3. Find the variance of \(X\).
4. One hundred observations of \(X\) are taken, and the mean of the observations is denoted by \(\bar { X }\). Write down the approximate distribution of \(\bar { X }\), giving the values of any parameters.
5. Write down the mean and the variance of the random variable \(Y\) with probability density function given by $$g ( y ) = \begin{cases} 4 \left( y + \frac { 4 } { 5 } \right) ^ { 3 } & - \frac { 4 } { 5 } \leqslant y \leqslant \frac { 1 } { 5 } \\ 0 & \text { otherwise } \end{cases}$$

3 The discrete random variable \(X\) has the following probability distribution The continuous random variable \(Y\) has the following probability density function $$\mathrm { f } ( y ) = \begin{cases} \frac { 1 } { 64 } y ^ { 3 } & 0 \leq y \leq 4 \\ 0 & \text { otherwise } \end{cases}$$ Given that \(X\) and \(Y\) are independent, show that \(\mathrm { E } \left( X ^ { 2 } + Y ^ { 2 } \right) = \frac { 1327 } { 60 }\)

4

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

2 The continuous random variable \(Y\) has probability density function \(\mathrm { f } ( y )\) where $$\int _ { - \infty } ^ { \infty } y \mathrm { f } ( y ) \mathrm { d } y = 16 \text { and } \int _ { - \infty } ^ { \infty } y ^ { 2 } \mathrm { f } ( y ) \mathrm { d } y = 1040$$ Find the standard deviation of \(Y\) Circle your answer.
[0pt] [1 mark]
28
32
784
1024

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

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.

8
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256 2 The random variable \(T\) has an exponential distribution with mean 2 Find \(\mathrm { P } ( T \leq 1.4 )\) Circle your answer. \(\mathrm { e } ^ { - 2.8 }\) \(\mathrm { e } ^ { - 0.7 }\) \(1 - e ^ { - 0.7 }\) \(1 - \mathrm { e } ^ { - 2.8 }\) The continuous random variable \(Y\) has cumulative distribution function $$\mathrm { F } ( y ) = \left\{ \begin{array} { l r } 0 & y < 2 \\ - \frac { 1 } { 9 } y ^ { 2 } + \frac { 10 } { 9 } y - \frac { 16 } { 9 } & 2 \leq y < 5 \\ 1 & y \geq 5 \end{array} \right.$$ Find the median of \(Y\) Circle your answer. 2 \(\frac { 10 - 3 \sqrt { 2 } } { 2 }\) \(\frac { 7 } { 2 }\) \(\frac { 10 + 3 \sqrt { 2 } } { 2 }\) Turn over for the next question 4 Research has shown that the mean number of volcanic eruptions on Earth each day is 20 Sandra records 162 volcanic eruptions during a period of one week. Sandra claims that there has been an increase in the mean number of volcanic eruptions per week. Test Sandra's claim at the \(5 \%\) level of significance.
5 The continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} \frac { 1 } { 6 } e ^ { \frac { x } { 3 } } & 0 \leq x \leq \ln 27 \\ 0 & \text { otherwise } \end{cases}$$ Show that the mean of \(X\) is \(\frac { 3 } { 2 } ( \ln 27 - 2 )\) 6 Over time it has been accepted that the mean retirement age for professional baseball players is 29.5 years old. Imran claims that the mean retirement age is no longer 29.5 years old.
He takes a random sample of 5 recently retired professional baseball players and records their retirement ages, \(x\). The results are $$\sum x = 152.1 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 7.81$$ 6

1. State an assumption that you should make about the distribution of the retirement ages to investigate Imran's claim. 6
2. Investigate Imran's claim, using the 10\% level of significance.

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256 2 The random variable \(T\) has an exponential distribution with mean 2 Find \(\mathrm { P } ( T \leq 1.4 )\) Circle your answer. \(\mathrm { e } ^ { - 2.8 }\) \(\mathrm { e } ^ { - 0.7 }\) \(1 - e ^ { - 0.7 }\) \(1 - \mathrm { e } ^ { - 2.8 }\) The continuous random variable \(Y\) has cumulative distribution function $$\mathrm { F } ( y ) = \left\{ \begin{array} { l r } 0 & y < 2 \\ - \frac { 1 } { 9 } y ^ { 2 } + \frac { 10 } { 9 } y - \frac { 16 } { 9 } & 2 \leq y < 5 \\ 1 & y \geq 5 \end{array} \right.$$ Find the median of \(Y\) Circle your answer. 2 \(\frac { 10 - 3 \sqrt { 2 } } { 2 }\) \(\frac { 7 } { 2 }\) \(\frac { 10 + 3 \sqrt { 2 } } { 2 }\) Turn over for the next question 4 Research has shown that the mean number of volcanic eruptions on Earth each day is 20 Sandra records 162 volcanic eruptions during a period of one week. Sandra claims that there has been an increase in the mean number of volcanic eruptions per week. Test Sandra's claim at the \(5 \%\) level of significance.
5 The continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} \frac { 1 } { 6 } e ^ { \frac { x } { 3 } } & 0 \leq x \leq \ln 27 \\ 0 & \text { otherwise } \end{cases}$$ Show that the mean of \(X\) is \(\frac { 3 } { 2 } ( \ln 27 - 2 )\) 6 Over time it has been accepted that the mean retirement age for professional baseball players is 29.5 years old. Imran claims that the mean retirement age is no longer 29.5 years old.
He takes a random sample of 5 recently retired professional baseball players and records their retirement ages, \(x\). The results are $$\sum x = 152.1 \quad \text { and } \quad \sum ( x - \bar { x } ) ^ { 2 } = 7.81$$ 6

1. State an assumption that you should make about the distribution of the retirement ages to investigate Imran's claim. 6
2. Investigate Imran's claim, using the 10\% level of significance.

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

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

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 random variable \(X\) has a uniform distribution on the interval \([ - 1,1 ]\), so that its probability density function is given by $$f ( x ) = \begin{cases} \frac { 1 } { 2 } & - 1 \leqslant x \leqslant 1 \\ 0 & \text { otherwise } \end{cases}$$

1. Show from the definition of the moment generating function that the moment generating function of \(X\) is \(\frac { \sinh t } { t }\).
2. By using the series expansion of \(\sinh t\), find the variance of \(X\) and the value of \(\mathrm { E } \left( X ^ { 4 } \right)\).

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

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