Continuous Uniform Random Variables

2. The random variable \(X\), which can take any value in the interval \(1 \leq X \leq n\), is modelled by the continuous uniform distribution with mean 12.

1. Show that \(n = 23\) and find the variance of \(X\).
2. Find \(\mathrm { P } ( 10 < X < 14 )\).

3 The continuous random variable \(R\) follows a rectangular distribution with probability density function given by $$f ( r ) = \begin{cases} k & - a \leq r \leq b
0 & \text { otherwise } \end{cases}$$ Prove, using integration, that \(\mathrm { E } ( R ) = \frac { 1 } { 2 } ( b - a )\)
[0pt] [4 marks]

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 \(X\).

3. A rectangle has a perimeter of 20 cm . The length, \(X \mathrm {~cm}\), of one side of this rectangle is uniformly distributed between 1 cm and 7 cm . Find the probability that the length of the longer side of the rectangle is more than 6 cm long.

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 \(100 \mathrm {~cm} ^ { 2 }\).

1 The continuous random variable \(X\) has probability density function $$f ( x ) = \begin{cases} \frac { 1 } { 5 } & 1 \leq x \leq 6
0 & \text { otherwise } \end{cases}$$ Find \(\mathrm { P } ( X \geq 3 )\)
Circle your answer. \(\frac { 1 } { 5 } \quad \frac { 2 } { 5 } \quad \frac { 3 } { 5 } \quad \frac { 4 } { 5 }\)

2. The continuous random variable \(X\) is uniformly distributed over the interval \([ 2,6 ]\).

1. Write down the probability density function \(\mathrm { f } ( x )\). Find
2. \(\mathrm { E } ( X )\),
3. \(\operatorname { Var } ( X )\),
4. the cumulative distribution function of \(X\), for all \(x\),
5. \(\mathrm { P } ( 2.3 < X < 3.4 )\).

3. The random variable \(X\) is uniformly distributed over the interval \([ - 1,5 ]\).

1. Sketch the probability density function \(\mathrm { f } ( x )\) of \(X\). Find
2. \(\mathrm { E } ( X )\),
3. \(\operatorname { Var } ( \mathrm { X } )\),
4. \(\mathrm { P } ( - 0.3 < X < 3.3 )\).

2. A continuous random variable \(X\) has the probability density function $$\begin{array} { l l } \mathrm { f } ( x ) = k & 5 \leq x \leq 15 ,
\mathrm { f } ( x ) = 0 & \text { otherwise. } \end{array}$$

1. Find \(k\) and specify the cumulative density function \(\mathrm { F } ( x )\).
2. Write down the value of \(\mathrm { P } ( X < 8 )\).

4. Jean catches a bus to work every morning. According to the timetable the bus is due at 8 a.m., but Jean knows that the bus can arrive at a random time between five minutes early and 9 minutes late. The random variable \(X\) represents the time, in minutes, after 7.55 a.m. when the bus arrives.

1. Suggest a suitable model for the distribution of \(X\) and specify it fully.
2. Calculate the mean time of arrival of the bus.
3. Find the cumulative distribution function of \(X\). Jean will be late for work if the bus arrives after 8.05 a.m.
4. Find the probability that Jean is late for work.

3. The continuous random variable \(X\) is uniformly distributed over the interval \([ - 1,3 ]\). Find

1. \(\mathrm { E } ( X )\)
2. \(\operatorname { Var } ( X )\)
3. \(\mathrm { E } \left( X ^ { 2 } \right)\)
4. \(\mathrm { P } ( X < 1.4 )\) A total of 40 observations of \(X\) are made.
5. Find the probability that at least 10 of these observations are negative.

6 A game consists of two rounds. The first round of the game uses a random number generator to output the score \(X\), a real number between 0 and 10 6

1. Find \(\mathrm { P } ( X > 4 )\) 6
2. The second round of the game uses an unbiased dice, with faces numbered 1 to 6 , to give the score \(Y\) The variables \(X\) and \(Y\) are independent.
   6
3. 1. Find the mean total score of the game.
      6
4. (ii) Find the variance of the total score of the game.

2. The random variable \(X\) follows a continuous uniform distribution over the interval \([ \alpha - 3,2 \alpha + 3 ]\) where \(\alpha\) is a constant.
The mean of a random sample of size \(n\) is denoted by \(\bar { X }\)

1. Show that \(\bar { X }\) is a biased estimator of \(\alpha\), and state the bias. Given that \(Y = k \bar { X }\) is an unbiased estimator for \(\alpha\)
2. find the value of \(k\). A random sample of 10 values of \(X\) is taken and the results are as follows $$\begin{array} { l l l l l l l l l l } 3 & 5 & 8 & 12 & 4 & 13 & 10 & 8 & 5 & 12 \end{array}$$
3. Hence estimate the maximum value of \(X\)

1. Helen believes that the random variable \(C\), representing cloud cover from the large data set, can be modelled by a discrete uniform distribution.
   1. Write down the probability distribution for \(C\).
   2. Using this model, find the probability that cloud cover is less than 50\%
   Helen used all the data from the large data set for Hurn in 2015 and found that the proportion of days with cloud cover of less than \(50 \%\) was 0.315
2. Comment on the suitability of Helen's model in the light of this information.
3. Suggest an appropriate refinement to Helen’s model.

1. The random variable \(R\) has a continuous uniform distribution over the interval [5,9]
   1. Specify fully the probability density function of \(R\).
   2. Find \(\mathrm { P } ( 7 < R < 10 )\)
   The random variable \(A\) is the area of a circle radius \(R \mathrm {~cm}\).
2. Find \(\mathrm { E } ( \mathrm { A } )\)

1. The time in minutes that Elaine takes to checkout at her local supermarket follows a continuous uniform distribution defined over the interval [3,9].

Find

1. Elaine's expected checkout time,
2. the variance of the time taken to checkout at the supermarket,
3. the probability that Elaine will take more than 7 minutes to checkout. Given that Elaine has already spent 4 minutes at the checkout,
4. find the probability that she will take a total of less than 6 minutes to checkout.

6. The three independent random variables \(A , B\) and \(C\) each has a continuous uniform distribution over the interval \([ 0,5 ]\).

1. Find \(\mathrm { P } ( A > 3 )\).
2. Find the probability that \(A , B\) and \(C\) are all greater than 3 . The random variable \(Y\) represents the maximum value of \(A , B\) and \(C\). The cumulative distribution function of \(Y\) is $$\mathrm { F } ( y ) = \begin{cases} 0 & y < 0
   \frac { y ^ { 3 } } { 125 } & 0 \leqslant y \leqslant 5
   1 & y > 5 \end{cases}$$
3. Find the probability density function of \(Y\).
4. Sketch the probability density function of \(Y\).
5. Write down the mode of \(Y\).
6. Find \(\mathrm { E } ( Y )\).
7. Find \(\mathrm { P } ( Y > 3 )\).

1
\includegraphics[max width=\textwidth, alt={}, center]{0cd5fc36-486d-4c24-b809-907b3e87cfd7-2_371_531_255_806} The diagram shows the graph of the probability density function, f , of a random variable \(X\). Find the median of \(X\).

1. (a) Explain the difference between a discrete and a continuous variable.

A random number generator on a calculator generates numbers, \(X\), to 3 decimal places, in the range 0 to 1 , e.g. 0.386 . The variable \(X\) may be modelled by a continuous uniform distribution, having the probability density function \(\mathrm { f } ( x )\), where $$\begin{array} { l l } \mathrm { f } ( x ) = 1 & 0 < x < 1
\mathrm { f } ( x ) = 0 & \text { otherwise } \end{array}$$ (b) Explain why this model is not totally accurate.
(c) Sketch the cumulative distribution function of \(X\).

8 The continuous random variable \(Y\) has a uniform distribution on [0,2].

1. It is given that \(\mathrm { E } [ a \cos ( a Y ) ] = 0.3\), where \(a\) is a constant between 0 and 1 , and \(a Y\) is measured in radians. Determine the value of the constant \(a\).
2. Determine the \(60 ^ { \text {th } }\) percentile of \(Y ^ { 2 }\).

1. The continuous random variable \(X\) is uniformly distributed over the interval [2, 7]
   1. Write down the value of \(\mathrm { E } ( X )\)
   2. Find \(\mathrm { P } ( 1 < X < 4 )\)
   3. Find \(\mathrm { P } \left( 2 X ^ { 2 } - 15 X + 27 > 0 \right)\)
   4. Find \(\mathrm { E } \left( \frac { 3 } { X ^ { 2 } } \right)\)