5.03f Relate pdf-cdf: medians and percentiles

A random variable \(X\) has probability density function given by $$f(x) = \begin{cases} k(1-x) & -1 \leq x \leq 1, \\ 0 & \text{otherwise}, \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac{1}{2}\). [2]
2. Find \(\text{P}(X > \frac{1}{2})\). [1]
3. Find the mean of \(X\). [3]
4. Find \(a\) such that \(\text{P}(X < a) = \frac{1}{3}\). [3]

The time, \(T\) minutes, taken by people to complete a test has probability density function given by $$\mathrm{f}(t) = \begin{cases} k(10t - t^2) & 5 \leq t \leq 10, \\ 0 & \text{otherwise}, \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac{3}{250}\). [3]
2. Find \(\mathrm{E}(T)\). [3]
3. Find the probability that a randomly chosen value of \(T\) lies between \(\mathrm{E}(T)\) and the median of \(T\). [3]
4. State the greatest possible length of time taken to complete the test. [1]

The average speed of a bus, \(x\) km h\(^{-1}\), on a certain journey is a continuous random variable \(X\) with probability density function given by $$\text{f}(x) = \begin{cases} \frac{k}{x^2} & 20 \leq x \leq 28, \\ 0 & \text{otherwise}. \end{cases}$$

1. Show that \(k = 70\). [3]
2. Find E\((X)\). [3]
3. Find P\((X < \text{E}(X))\). [2]
4. Hence determine whether the mean is greater or less than the median. [2]

The continuous random variable \(X\) has probability density function f given by $$f(x) = \begin{cases} \frac{1}{8} & 0 \leq x < 1, \\ \frac{1}{28}(8 - x) & 1 \leq x \leq 8, \\ 0 & \text{otherwise}. \end{cases}$$

1. Find the cumulative distribution function of \(X\). [3]

1. Find the value of the constant \(a\) such that P\((X \leq a) = \frac{5}{7}\). [3]

The random variable \(Y\) is given by \(Y = \sqrt[3]{X}\).

1. Find the probability density function of \(Y\). [5]

A continuous random variable \(X\) has cumulative distribution function $$\mathrm{F}(x) = \begin{cases} 0 & x < 0 \\ \frac{1}{4}x & 0 \leq x \leq 1 \\ \frac{1}{20}x^4 + \frac{1}{5} & 1 < x \leq d \\ 1 & x > d \end{cases}$$

1. Show that \(d = 2\) [2]
2. Find \(\mathrm{P}(X < 1.5)\) [2]
3. Write down the value of the lower quartile of \(X\) [1]
4. Find the median of \(X\) [3]
5. Find, to 3 significant figures, the value of \(k\) such that \(\mathrm{P}(X > 1.9) = \mathrm{P}(X < k)\) [4]

The continuous random variable X has cumulative distribution function F(x) given by $$\text{F}(x) = \begin{cases} 0, & x < 1 \\ \frac{1}{2}(-x^3 + 6x^2 - 5), & 1 \leq x \leq 4 \\ 1, & x > 4 \end{cases}$$

1. Find the probability density function f(x). [3]
2. Find the mode of X. [2]
3. Sketch f(x) for all values of x. [3]
4. Find the mean \(\mu\) of X. [3]
5. Show that F(\(\mu\)) > 0.5. [1]
6. Show that the median of X lies between the mode and the mean. [2]

A drinks machine dispenses lemonade into cups. It is electronically controlled to cut off the flow of lemonade randomly between 180 ml and 200 ml. The random variable X is the volume of lemonade dispensed into a cup.

1. Specify the probability density function of X and sketch its graph. [4]

Find the probability that the machine dispenses

1. less than 183 ml, [3]
2. exactly 183 ml. [1]
3. Calculate the inter-quartile range of X. [3]
4. Determine the value of s such that P(X ≤ s) = 1 - 2P(X ≤ s). [2]
5. Interpret in words your value of s.

A random variable \(X\) has probability density function given by $$\text{f}(x) = \begin{cases} -\frac{2}{9}x + \frac{8}{9} & 1 \leqslant x \leqslant 4 \\ 0 & \text{otherwise} \end{cases}$$

1. Show that the cumulative distribution function F(x) can be written in the form \(ax^2 + bx + c\), for \(1 \leqslant x \leqslant 4\) where \(a\), \(b\) and \(c\) are constants. [3]
2. Define fully the cumulative distribution function F(x). [2]
3. Show that the upper quartile of \(X\) is 2.5 and find the lower quartile. [6]

Given that the median of \(X\) is 1.88

1. describe the skewness of the distribution. Give a reason for your answer. [2]

A continuous random variable \(X\) has the probability density function f(\(x\)) shown in Figure 1. \includegraphics{figure_1} Figure 1

1. Show that f(\(x\)) = \(4 - 8x\) for \(0 \leqslant x \leqslant 0.5\) and specify f(\(x\)) for all real values of \(x\). [4]
2. Find the cumulative distribution function F(\(x\)). [4]
3. Find the median of \(X\). [3]
4. Write down the mode of \(X\). [1]
5. State, with a reason, the skewness of \(X\). [1]

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 lifetime, \(X\), in tens of hours, of a battery has a cumulative distribution function F(x) given by $$\text{F}(x) = \begin{cases} 0 & x < 1 \\ \frac{4}{9}(x^2 + 2x - 3) & 1 \leqslant x \leqslant 1.5 \\ 1 & x > 1.5 \end{cases}$$

1. Find the median of \(X\), giving your answer to 3 significant figures. [3]
2. Find, in full, the probability density function of the random variable \(X\). [3]
3. Find P(\(X \geqslant 1.2\)) [2]

A camping lantern runs on 4 batteries, all of which must be working. Four new batteries are put into the lantern.

1. Find the probability that the lantern will still be working after 12 hours. [2]

A random variable \(X\) has probability density function given by $$f(x) = \begin{cases} kx^2 & 0 \leq x \leq 2 \\ k\left(1 - \frac{x}{6}\right) & 2 < x \leq 6 \\ 0 & \text{otherwise} \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac{1}{4}\) [4]
2. Write down the mode of \(X\). [1]
3. Specify fully the cumulative distribution function F(\(x\)). [5]
4. Find the upper quartile of \(X\). [4]

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 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]

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]

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 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]

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 \(X\) 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]