5.03a Continuous random variables: pdf and cdf

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]

The random variable \(X\) has probability density function defined by $$f(x) = \begin{cases} \frac{1}{2} & 0 \leqslant x \leqslant 1 \\ \frac{1}{18}(x - 4)^2 & 1 \leqslant x \leqslant 4 \\ 0 & \text{otherwise} \end{cases}$$

1. State values for the median and the lower quartile of \(X\). [2 marks]
2. Show that, for \(1 \leqslant x \leqslant 4\), the cumulative distribution function, \(\mathrm{F}(x)\), of \(X\) is given by $$\mathrm{F}(x) = 1 + \frac{1}{54}(x - 4)^3$$ (You may assume that \(\int (x - 4)^2 \, dx = \frac{1}{3}(x - 4)^3 + c\).) [4 marks]
3. Determine \(\mathrm{P}(2 \leqslant X \leqslant 3)\). [2 marks]
4. 1. Show that \(q\), the upper quartile of \(X\), satisfies the equation \((q - 4)^3 = -13.5\). [3 marks]
   2. Hence evaluate \(q\) to three decimal places. [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]

The continuous random variable \(X\) has a cumulative distribution function F(\(x\)), where $$\text{F}(x) = \begin{cases} 0 & x < 1 \\ \frac{1}{4}(x - 1) & 1 \leqslant x < 4 \\ \frac{1}{16}(12x - x^2 - 20) & 4 \leqslant x \leqslant 6 \\ 1 & x > 6 \end{cases}$$

1. Sketch the probability density function, f(\(x\)), on the grid below. [5 marks]
2. Find the mean value of \(X\). [4 marks]

Light bulbs produced in a certain factory have lifetimes, in 100s of hours, whose distribution is modelled by the random variable \(X\) with probability density function $$f(x) = \frac{2x(3-x)}{9}, \quad 0 \leq x \leq 3;$$ $$f(x) = 0 \quad \text{otherwise}.$$

1. Sketch \(f(x)\). [2 marks]
2. Write down the mean lifetime of a bulb. [1 mark]
3. Show that ten times as many bulbs fail before 200 hours as survive beyond 250 hours. [5 marks]
4. Given that a bulb lasts for 200 hours, find the probability that it will then last for at least another 50 hours. [2 marks]
5. State, with a reason, whether you consider that the density function \(f\) is a realistic model for the lifetimes of light bulbs. [1 mark]

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]

The waiting time, in minutes, at a dentist is modelled by the continuous random variable \(T\) with probability density function $$f(t) = k(10 - t) \quad 0 \leq t \leq 10,$$ $$f(t) = 0 \quad \text{otherwise}.$$

1. Sketch the graph of \(f(t)\) and find the value of \(k\). [4 marks]
2. Find the mean value of \(T\). [4 marks]
3. Find the 95th percentile of \(T\). [3 marks]
4. State whether you consider this function to be a sensible model for \(T\) and suggest how it could be modified to provide a better model. [2 marks]

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 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 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]

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]

A continuous random variable \(X\) has the cumulative distribution function $$F(x) = 0 \quad x < 2,$$ $$F(x) = k(x - a)^2 \quad 2 \leq x \leq 6,$$ $$F(x) = 1 \quad x \geq 6.$$

1. Find the values of the constants \(a\) and \(k\). [4 marks]
2. Show that the median of \(X\) is \(2(1 + \sqrt{2})\). [4 marks]
3. Given that \(X > 4\), find the probability that \(X > 5\). [6 marks]

A continuous random variable \(X\) has the probability density function $$\text{f}(x) = \frac{6x}{175} \quad 0 \leq x < 5,$$ $$\text{f}(x) = \frac{6x(10-x)}{875} \quad 5 \leq x \leq 10,$$ $$\text{f}(x) = 0 \quad \text{otherwise}.$$

1. Verify that f is a probability density function. [6 marks]
2. Write down the probability that \(X < 1\). [2 marks]
3. Find the cumulative distribution function of \(X\), carefully showing how it changes for different domains. [7 marks]
4. Find the probability that \(2 < X < 7\). [2 marks]

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]

A golfer believes that the distance, in metres, that she hits a ball with a 5 iron, follows a continuous uniform distribution over the interval \([100, 150]\).

1. Find the median and interquartile range of the distance she hits a ball, that would be predicted by this model. [3 marks]
2. Explain why the continuous uniform distribution may not be a suitable model. [2 marks]

The continuous random variable \(X\) has the following cumulative distribution function: $$F(x) = \begin{cases} 0, & x < 0, \\ \frac{1}{64}(16x - x^2), & 0 \leq x \leq 8, \\ 1, & x > 8. \end{cases}$$

1. Find \(P(X > 5)\). [2 marks]
2. Find and specify fully the probability density function \(f(x)\) of \(X\). [3 marks]
3. Sketch \(f(x)\) for all values of \(x\). [3 marks]

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 continuous random variable \(X\) has the following probability density function: $$f(x) = \begin{cases} \frac{1}{8}x, & 0 \leq x \leq 2, \\ \frac{1}{12}(6-x), & 2 \leq x \leq 6, \\ 0, & \text{otherwise}. \end{cases}$$

1. Sketch \(f(x)\) for all values of \(x\). [4 marks]
2. State the mode of \(X\). [1 mark]
3. Define fully the cumulative distribution function \(F(x)\) of \(X\). [9 marks]
4. Show that the median of \(X\) is 2.536, correct to 4 significant figures. [4 marks]

The continuous random variable \(X\) has the following cumulative distribution function: $$\text{F}(x) = \begin{cases} 0, & x < 0, \\ \frac{1}{432} x^2(x^2 - 16x + 72), & 0 \leq x \leq 6, \\ 1, & x > 6. \end{cases}$$

1. Find P(\(X < 2\)). [2 marks]
2. Find and specify fully the probability density function f(\(x\)) of \(X\). [4 marks]
3. Show that the mode of \(X\) is 2. [6 marks]
4. State, with a reason, whether the median of \(X\) is higher or lower than the mode of \(X\). [1 mark]

A class of children are each asked to draw a line that they think is 10 cm long without using a ruler. The teacher models how many centimetres each child's line is longer than 10 cm by the random variable \(X\) and believes that \(X\) has the following probability density function: $$f(x) = \begin{cases} \frac{1}{8}, & -4 \leq x \leq 4, \\ 0, & \text{otherwise}. \end{cases}$$

1. Write down the name of this distribution. [1]
2. Define fully the cumulative distribution function F(x) of \(X\). [4]
3. Calculate the proportion of children making an error of less than 15\% according to this model. [3]
4. Give two reasons why this may not be a very suitable model. [2]

The length of time, in tens of minutes, that patients spend waiting at a doctor's surgery is modelled by the continuous random variable \(T\), with the following cumulative distribution function: $$F(t) = \begin{cases} 0, & t < 0, \\ \frac{1}{135}(54t + 9t^2 - 4t^3), & 0 \leq t \leq 3, \\ 1, & t > 3. \end{cases}$$

1. Find the probability that a patient waits for more than 20 minutes. [3]
2. Show that the median waiting time is between 11 and 12 minutes. [3]
3. Define fully the probability density function f(t) of \(T\). [3]
4. Find the modal waiting time in minutes. [4]
5. Give one reason why this model may need to be refined. [1]