5.03c Calculate mean/variance: by integration

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 queuing time in minutes, \(X\), of a customer at a post office is modelled by the probability density function $$\text{f}(x) = \begin{cases} kx(81 - x^2) & 0 \leqslant x \leqslant 9 \\ 0 & \text{otherwise} \end{cases}$$

1. Show that \(k = \frac{4}{6561}\). [3]

Using integration, find

1. the mean queuing time of a customer, [4]
2. the probability that a customer will queue for more than 5 minutes. [3]

Three independent customers shop at the post office.

1. Find the probability that at least 2 of the customers queue for more than 5 minutes. [3]

The continuous random variable \(X\) has probability density function $$f(x) = \begin{cases} \frac{x}{15}, & 0 \leq x \leq 2, \\ \frac{2}{15}, & 2 < x < 7, \\ \frac{4}{9} - \frac{2x}{45}, & 7 \leq x \leq 10, \\ 0, & \text{otherwise}. \end{cases}$$

1. Sketch \(f(x)\) for all values of \(x\). [3]
2. 1. Find expressions for the cumulative distribution function, \(\mathrm{F}(x)\), for \(0 \leq x \leq 2\) and for \(7 \leq x \leq 10\).
   2. Show that for \(2 < x < 7\), \(\mathrm{F}(x) = \frac{2x}{15} - \frac{2}{15}\).
   3. Specify \(\mathrm{F}(x)\) for \(x < 0\) and for \(x > 10\).
   [8]
3. Find \(\mathrm{P}(X \leq 8.2)\). [2]
4. Find, to 3 significant figures, \(\mathrm{E}(X)\). [4]

A continuous random variable \(X\) has probability density function f(\(x\)) where $$\text{f}(x) = \begin{cases} k(x^2 + 2x + 1) & -1 \leq x \leq 0, \\ 0, & \text{otherwise} \end{cases}$$ where \(k\) is a positive integer.

1. Show that \(k = 3\). [4]

Find

1. E(\(X\)), [4]
2. the cumulative distribution function F(\(x\)), [4]
3. P(\(-0.3 < X < 0.3\)). [3]

A random variable \(X\) has probability density function given by $$f(x) = \begin{cases} \frac{1}{3}, & 0 \leq x \leq 1, \\ \frac{8x^3}{45}, & 1 \leq x \leq 2, \\ 0, & \text{otherwise}. \end{cases}$$

1. Calculate the mean of \(X\). [5]
2. Specify fully the cumulative distribution function F\((x)\). [7]
3. Find the median of \(X\). [3]
4. Comment on the skewness of the distribution of \(X\). [2]

The continuous random variable \(X\) has probability density function $$f(x) = \begin{cases} \frac{1+x}{k}, & 1 \leqslant x \leqslant 4, \\ 0, & \text{otherwise}. \end{cases}$$

1. Show that \(k = \frac{21}{2}\). [3]
2. Specify fully the cumulative distribution function of \(X\). [5]
3. Calculate E\((X)\). [3]
4. Find the value of the median. [3]
5. Write down the mode. [1]
6. Explain why the distribution is negatively skewed. [1]

The random variable \(Y\) has probability density function f(y) given by $$\text{f}(y) = \begin{cases} ky(a - y) & 0 \leqslant y \leqslant 3 \\ 0 & \text{otherwise} \end{cases}$$ where \(k\) and \(a\) are positive constants.

1. 1. Explain why \(a \geqslant 3\)
   2. Show that \(k = \frac{2}{9(a-2)}\)
   [6]

Given that E(Y) = 1.75

1. show that \(a = 4\) and write down the value of \(k\). [6]

For these values of \(a\) and \(k\),

1. sketch the probability density function, [2]
2. write down the mode of \(Y\). [1]

The continuous random variable \(L\) represents the error, in metres, made when a machine cuts poles to a target length. The distribution of \(L\) is a continuous uniform distribution over the interval [0, 0.5]

1. Find P(\(L < 0.4\)). [1]
2. Write down E(\(L\)). [1]
3. Calculate Var(\(L\)). [2]

A random sample of 30 poles cut by this machine is taken.

1. Find the probability that fewer than 4 poles have an error of more than 0.4 metres from the target length. [3]

When a new machine cuts poles to a target length, the error, \(X\) metres, is modelled by the cumulative distribution function F(\(x\)) where $$\text{F}(x) = \begin{cases} 0 & x < 0 \\ 4x - 4x^2 & 0 \leq x \leq 0.5 \\ 1 & \text{otherwise} \end{cases}$$

1. Using this model, find P(\(X > 0.4\)) [2]

A random sample of 100 poles cut by this new machine is taken.

1. Using a suitable approximation, find the probability that at least 8 of these poles have an error of more than 0.4 metres. [3]

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

1. Find \(k\) in terms of \(n\). [3]
2. Find E(\(X\)) in terms of \(n\). [3]
3. Find E(\(X^2\)) in terms of \(n\). [2]

Given that \(n = 2\)

1. find Var(3\(X\)). [3]

The continuous random variable \(X\) has probability density function f(\(x\)) given by $$\text{f}(x) = \begin{cases} \frac{1}{20}x^3, & 1 \leq x \leq 3 \\ 0, & \text{otherwise} \end{cases}$$

1. Sketch f(\(x\)) for all values of \(x\). [3]
2. Calculate E(\(X\)). [3]
3. Show that the standard deviation of \(X\) is 0.459 to 3 decimal places. [3]
4. Show that for \(1 \leq x \leq 3\), P(\(X \leq x\)) is given by \(\frac{1}{80}(x^4 - 1)\) and specify fully the cumulative distribution function of \(X\). [5]
5. Find the interquartile range for the random variable \(X\). [4]

Some statisticians use the following formula to estimate the interquartile range: $$\text{interquartile range} = \frac{4}{3} \times \text{standard deviation}.$$

1. Use this formula to estimate the interquartile range in this case, and comment. [2]

The continuous random variable \(X\) has a rectangular distribution defined by $$f(x) = \begin{cases} k & -3k \leqslant x \leqslant k \\ 0 & \text{otherwise} \end{cases}$$

1. 1. Sketch the graph of f. [2 marks]
   2. Hence show that \(k = \frac{1}{2}\). [2 marks]
2. Find the exact numerical values for the mean and the standard deviation of \(X\). [3 marks]
3. 1. Find \(\mathrm{P}\left(X \geqslant -\frac{1}{4}\right)\). [2 marks]
   2. Write down the value of \(\mathrm{P}\left(X \neq -\frac{1}{4}\right)\). [1 mark]

A digital thermometer measures temperatures in degrees Celsius. The thermometer rounds down the actual temperature to one decimal place, so that, for example, 36.23 and 36.28 are both shown as 36.2. The error, \(X\) °C, resulting from this rounding down can be modelled by a rectangular distribution with the following probability density function. $$f(x) = \begin{cases} k & 0 \leqslant x \leqslant 0.1 \\ 0 & \text{otherwise} \end{cases}$$

1. State the value of \(k\). [1 mark]
2. Find the probability that the error resulting from this rounding down is greater than 0.03 °C. [1 mark]
3. 1. State the value for E(\(X\)).
   2. Use integration to find the value for E(\(X^2\)).
   3. Hence find the value for the standard deviation of \(X\).
   [5 marks]

Some children are asked to mark the centre of a scale 10 cm long. The position they choose is indicated by the variable \(X\), where \(0 \leq X \leq 10\). Initially, \(X\) is modelled as a random variable with a continuous uniform distribution.

1. Find the mean and the standard deviation of \(X\). [3 marks]

It is suggested that a better model would be the distribution with probability density function $$f(x) = cx, \quad 0 \leq x \leq 5, \quad f(x) = c(10-x), \quad 5 < x \leq 10, \quad f(x) = 0 \text{ otherwise}.$$

1. Write down the mean of \(X\). [1 mark]
2. Find \(c\), and hence find the standard deviation of \(X\) in this model. [7 marks]
3. Find P(\(4 < X < 6\)). [3 marks]

It is then proposed that an even better model for \(X\) would be a Normal distribution with the mean and standard deviation found in parts (b) and (c).

1. Use these results to find P(\(4 < X < 6\)) in the third model. [4 marks]
2. Compare your answer with (d). Which model do you think is most appropriate? [1 mark]

A continuous random variable \(X\) has a probability density function given by $$f(x) = \frac{x^2}{312} \quad 4 \leq x \leq 10,$$ $$f(x) = 0 \quad \text{otherwise}.$$

1. Find E\((X)\). [3 marks]
2. Find the variance of \(X\). [4 marks]
3. Find the cumulative distribution function F\((x)\), for all values of \(x\). [5 marks]
4. Hence find the median value of \(X\). [3 marks]
5. Write down the modal value of \(X\). [1 mark]

It is sometimes suggested that, for most distributions, $$2 \times (\text{median} - \text{mean}) \approx \text{mode} - \text{median}.$$

1. Show that this result is not satisfied in this case, and suggest a reason why. [2 marks]

A random variable \(X\) has a probability density function given by $$f(x) = \frac{4x^2(3-x)}{27} \quad 0 \leq x \leq 3,$$ $$f(x) = 0 \quad \text{otherwise}.$$

1. Find the mode of \(X\). [3 marks]
2. Find the mean of \(X\). [3 marks]
3. Specify completely the cumulative distribution function of \(X\). [4 marks]
4. Deduce that the median, \(m\), of \(X\) satisfies the equation \(m^4 - 4m^3 + 13·5 = 0\), and hence show that \(1·84 < m < 1·85\). [4 marks]
5. What do these results suggest about the skewness of the distribution? [1 mark]

A corner-shop has weekly sales (in thousands of pounds), which can be modelled by the continuous random variable \(X\) with probability density function $$f(x) = k(x-2)(10-x) \quad 2 \leq x \leq 10,$$ $$f(x) = 0 \quad \text{otherwise}.$$

1. Show that \(k = \frac{3}{256}\) and write down the mean of \(X\). [6 marks]
2. Find the standard deviation of the weekly sales. [6 marks]
3. Find the probability that the sales exceed £8 000 in any particular week. [4 marks]

If the sales exceed £8 000 per week for 4 consecutive weeks, the manager gets a bonus.

1. Find the probability that the manager gets a bonus in February. [2 marks]

A continuous random variable \(X\) has probability density function f(x) given by $$\text{f(x)} = \frac{2x}{3} \quad 0 \leq x < 1,$$ $$\text{f(x)} = 1 - \frac{x}{3} \quad 1 \leq x \leq 3,$$ $$\text{f(x)} = 0 \quad \text{otherwise}.$$

1. Sketch the graph of f(x) for all \(x\). [3 marks]
2. Find the mean of \(X\). [5 marks]
3. Find the standard deviation of \(X\). [7 marks]
4. Show that the cumulative distribution function of \(X\) is given by $$\text{F(x)} = \frac{x^2}{3} \quad 0 \leq x < 1,$$ and find F(x) for \(1 \leq x \leq 3\). [6 marks]

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]

Two people are playing darts. Peg hits points randomly on the circular board, whose radius is \(a\). If the distance from the centre \(O\) of the point that she hits is modelled by the variable \(R\),

1. explain why the cumulative distribution function \(F(r)\) is given by $$F(r) = 0 \quad r < 0,$$ $$F(r) = \frac{r^2}{a^2} \quad 0 \leq r \leq a,$$ $$F(r) = 1 \quad r > a.$$ [4 marks]
2. By first finding the probability density function of \(R\), show that the mean distance from \(O\) of the points that Peg hits is \(\frac{2a}{3}\). [7 marks] Bob, a more experienced player, aims for \(O\), and his points have a distance \(X\) from \(O\) whose cumulative distribution function is $$F(x) = 0, \quad x < 0; \quad F(x) = \frac{x}{a}\left(2 - \frac{x}{a}\right), \quad 0 \leq x \leq a; \quad F(x) = 1, \quad x > a.$$
3. Find the probability density function of \(X\), and explain why it shows that Bob is aiming for \(O\). [5 marks]

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]

A continuous random variable \(X\) has probability density function $$\text{f}(x) = \begin{cases} ax^{-3} + bx^{-4} & x \geq 1, \\ 0 & \text{otherwise,} \end{cases}$$ where \(a\) and \(b\) are constants.

1. Explain what the letter \(x\) represents. [1]

It is given that P\((X > 2) = \frac{3}{16}\).

1. Show that \(a = 1\), and find the value of \(b\). [7]
2. Find E\((X)\). [3]

The continuous random variable \(T\) has the following probability density function: $$f(t) = \begin{cases} k(t^2 + 2), & 0 \leq t \leq 3, \\ 0, & \text{otherwise}. \end{cases}$$

1. Show that \(k = \frac{1}{15}\). [4 marks]
2. Sketch \(f(t)\) for all values of \(t\). [3 marks]
3. State the mode of \(T\). [1 mark]
4. Find \(E(T)\). [5 marks]
5. Show that the standard deviation of \(T\) is 0.798 correct to 3 significant figures. [6 marks]

In a test studying reaction times, white dots appear at random on a black rectangular screen. The continuous random variable \(X\) represents the distance, in centimetres, of the dot from the left-hand edge of the screen. The distribution of \(X\) is rectangular over the interval \([0, 20]\).

1. Find \(P(2 < X < 3.6)\). [2 marks]
2. Find the mean and variance of \(X\). [3 marks]

The continuous random variable \(Y\) represents the distance, in centimetres, of the dot from the bottom edge of the screen. The distribution of \(Y\) is rectangular over the interval \([0, 16]\). Find the probability that a dot appears

1. in a square of side 4 cm at the centre of the screen, [4 marks]
2. within 2 cm of the edge of the screen. [4 marks]

The random variable \(X\) follows a continuous uniform distribution over the interval \([2, 11]\).

1. Write down the mean of \(X\). [1 mark]
2. Find P(\(X \geq 8.6\)). [2 marks]
3. Find P(\(|X - 5| < 2\)). [2 marks]

The random variable \(Y\) follows a continuous uniform distribution over the interval \([a, b]\).

1. Show by integration that $$\text{E}(Y^2) = \frac{1}{3}(b^2 + ab + a^2).$$ [5 marks]
2. Hence, prove that $$\text{Var}(Y) = \frac{1}{12}(b - a)^2.$$ You may assume that E(\(Y\)) = \(\frac{1}{2}(a + b)\). [4 marks]