5.03b Solve problems: using pdf

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 continuous random variable X has probability density function f(x) where $$f(x) = \begin{cases} k(x^3 + 2x + 1), & -1 \leq x \leq 0, \\ 0, & otherwise \end{cases}$$ 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]

The continuous random variable \(X\) has probability density function $$\text{f}(x) = \begin{cases} kx(5 - x), & 0 \leq x \leq 4, \\ 0, & \text{otherwise,} \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac{3}{56}\). [3]
2. Find the cumulative distribution function F(\(x\)) for all values of \(x\). [4]
3. Evaluate E(\(X\)). [3]
4. Find the modal value of \(X\). [3]
5. Verify that the median value of \(X\) lies between 2.3 and 2.5. [3]
6. Comment on the skewness of \(X\). Justify your answer. [2]

The continuous random variable \(X\) is uniformly distributed over the interval \([-2, 7]\).

1. Write down fully the probability density function f(x) of \(X\). [2]
2. Sketch the probability density function f(x) of \(X\). [2]

Find

1. E(\(X^2\)), [3]
2. P(\(-0.2 < X < 0.6\)). [2]

The length of a telephone call made to a company is denoted by the continuous random variable \(T\). It is modelled by the probability density function $$\text{f}(t) = \begin{cases} kt & 0 \leqslant t \leqslant 10 \\ 0 & \text{otherwise} \end{cases}$$

1. Show that the value of \(k\) is \(\frac{1}{50}\). [3]
2. Find P(\(T > 6\)). [2]
3. Calculate an exact value for E(\(T\)) and for Var(\(T\)). [5]
4. Write down the mode of the distribution of \(T\). [1]

It is suggested that the probability density function, f(\(t\)), is not a good model for \(T\).

1. Sketch the graph of a more suitable probability density function for \(T\). [1]

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 \(R\) is uniformly distributed on the interval \(\alpha \leq R \leq \beta\). Given that \(\mathrm{E}(R) = 3\) and \(\mathrm{Var}(R) = \frac{25}{3}\), find

1. the value of \(\alpha\) and the value of \(\beta\), [7]
2. \(\mathrm{P}(R < 6.6)\). [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]
2. Find the probability that the machine dispenses
   1. less than 183 ml,
   2. exactly 183 ml.
   [3]
3. Calculate the inter-quartile range of \(X\). [1]
4. Determine the value of \(x\) such that P(\(X \geq x\)) = 2P(\(X \leq x\)). [3]
5. Interpret in words your value of \(x\). [2]

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]

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]

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 \(L\) 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]

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]