Continuous Uniform Random Variables

The continuous random variable R is uniformly distributed on the interval \(\alpha \leq R \leq \beta\). Given that E(R) = 3 and Var(R) = \(\frac{4}{3}\), find

1. the value of \(\alpha\) and the value of \(\beta\), [7]
2. P(R < 6.6). [2]

2. The continuous random variable \(X\) is uniformly distributed over the interval \([ \alpha , \beta ]\) where \(\beta > \alpha\) Given that \(\mathrm { E } ( X ) = 8\)

1. write down an equation involving \(\alpha\) and \(\beta\) Given also that \(\mathrm { P } ( X \leqslant 13 ) = 0.7\)
2. find the value of \(\alpha\) and the value of \(\beta\)
3. find \(\operatorname { Var } ( X )\)
4. find \(\mathrm { P } ( 5 \leqslant X \leqslant 35 )\)

1. The continuous random variable \(X\) is uniformly distributed over the interval \([ a , b ]\) where \(0 < a < b\)

Given that \(\mathrm { P } ( X < b - 2 a ) = \frac { 1 } { 3 }\)

1. 1. show that \(\mathrm { E } ( X ) = \frac { 5 a } { 2 }\)
   2. find \(\mathrm { P } ( X > b - 4 a )\) The continuous random variable \(Y\) is uniformly distributed over the interval [3, c] where \(c > 3\) Given that \(\operatorname { Var } ( Y ) = 3 c - 9\), find
2. 1. the value of \(c\)
   2. \(\mathrm { P } ( 2 Y - 7 < 20 - Y )\)
   3. \(\mathrm { E } \left( Y ^ { 2 } \right)\)

A child cuts a 30 cm piece of string into two parts, cutting at a random point.

1. Name the distribution of \(L\), the length of the longer part of string, and sketch the probability density function for \(L\). [4 marks]
2. Find the probability that one part of the string is more than twice as long as the other. [3 marks]

A piece of string \(AB\) has length 12 cm. A child cuts the string at a randomly chosen point \(P\), into two pieces. The random variable \(X\) represents the length, in cm, of the piece \(AP\).

1. Suggest a suitable model for the distribution of \(X\) and specify it fully [2]
2. Find the cumulative distribution function of \(X\). [4]
3. Write down P(\(X < 4\)). [1]

1. The random variable \(W\) has a continuous uniform distribution over the interval \([ - 6 , a ]\) where \(a\) is a constant.

Given that \(\operatorname { Var } ( W ) = 27\)

1. show that \(a = 12\) Given that \(\mathrm { P } ( W > b ) = \frac { 3 } { 5 }\)
2. 1. find the value of \(b\)
   2. find \(\mathrm { P } \left( - 12 < W < \frac { b } { 2 } \right)\) A piece of wood \(A B\) has length 160 cm . The wood is cut at random into 2 pieces. Each of the pieces is then cut in half. The four pieces are used to form the sides of a rectangle.
3. Calculate the probability that the area of the rectangle is greater than \(975 \mathrm {~cm} ^ { 2 }\)

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 \(\mathrm { P } ( X < 8 )\).

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]

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. \(\mathrm { P } ( 2.3 < X < 3.4 )\).

1. An engineer measures, to the nearest cm , the lengths of metal rods.
   1. Suggest a suitable model to represent the difference between the true lengths and the measured lengths.
   2. Find the probability that for a randomly chosen rod the measured length will be within 0.2 cm of the true length.
   Two rods are chosen at random.
2. Find the probability that for both rods the measured lengths will be within 0.2 cm of their true lengths.

A rectangle has a perimeter of 20 cm. The length, \(x\) 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. [5]

1. Navtej travels to work by train. A train leaves the station every 7 minutes and Navtej's arrival at the station is independent of when the train is due to leave.
   1. Write down a suitable model for the distribution of the time, \(T\) minutes, that he has to wait for a train to leave.
   2. Find the mean and standard deviation of \(T\)
   During a 10-week period, Navtej travels to work by train on 46 occasions.
2. Estimate the probability that the mean length of time that he has to wait for a train to leave is between 3.4 and 3.6 minutes.
3. State a necessary assumption for the calculation in part (c).

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

1. E(\(X\)) [1]
2. Var(\(X\)) [2]
3. E(\(X^2\)) [2]
4. P(\(X < 1.4\)) [1]

A total of 40 observations of \(X\) are made.

1. Find the probability that at least 10 of these observations are negative. [5]

The continuous random variable \(X\) is uniformly distributed over the interval \([-1, 4]\). Find

1. P\((X < 2.7)\), [1]
2. E\((X)\), [2]
3. Var \((X)\). [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. \(\frac { 1 } { 5 } \quad \frac { 2 } { 5 } \quad \frac { 3 } { 5 } \quad \frac { 4 } { 5 }\)

The continuous random variable \(T\) is equally likely to take any value from 5.0 to 11.0 inclusive.

1. Sketch the graph of the probability density function of \(T\). [2]
2. Write down the value of E(\(T\)) and find by integration the value of Var(\(T\)). [5]
3. A random sample of 48 observations of \(T\) is obtained. Find the approximate probability that the mean of the sample is greater than 8.3, and explain why the answer is an approximation. [6]

In this question you must show detailed reasoning. An ant starts from a fixed point \(O\) and walks in a straight line for \(1.5\) s. Its velocity, \(v\) cms\(^{-1}\), can be modelled by \(v = \frac{1}{\sqrt{9-t^2}}\). By finding the mean value of \(v\) in \(0 \leq t \leq 1.5\), deduce the average velocity of the ant. [5]

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 (b) (i) Find the mean total score of the game.
   6 (b) (ii) Find the variance of the total score of the game.

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 \(X\).

The random variable \(X\) has a continuous uniform distribution on the interval \(a \leq X \leq 3a\).

1. Without assuming any standard results, prove that \(\mu\), the mean value of \(X\), is equal to \(2a\) and derive an expression for \(\sigma^2\), the variance of \(X\), in terms of \(a\). [7 marks]
2. Find the probability that \(|X - \mu| < \sigma\) and compare this with the same probability when \(x\) is modelled by a Normal distribution with the same mean and variance. [6 marks]

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. \(\mathrm { P } ( - 0.3 < X < 3.3 )\).

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.

1. The random variable \(G\) has a continuous uniform distribution over the interval \([ - 3,15 ]\)
   1. Calculate \(\mathrm { P } ( G > 12 )\)
   The random variable \(H\) has a continuous uniform distribution over the interval [2, w] The random variables \(G\) and \(H\) are independent and \(\mathrm { E } ( H ) = 10\)
2. Show that the probability that \(G\) and \(H\) are both greater than 12 is \(\frac { 1 } { 16 }\) The random variable \(A\) is the area on a coordinate grid bounded by $$\begin{aligned} & y = - 3 \\ & y = - 4 | x | + k \end{aligned}$$ where \(k\) is a value from the continuous uniform distribution over the interval [5,10]
3. Calculate the expected value of \(A\)

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 \(\mathrm { E } \left( \frac { 3 } { X ^ { 2 } } \right)\)

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 \(Y ^ { 2 }\).