5.03b Solve problems: using pdf

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]

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]

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]

A railway company is investigating operations at a junction where delays often occur. Delays (in minutes) are modelled by the random variable \(T\) with the following cumulative distribution function. $$F(t) = \begin{cases} 0 & t \leq 0 \\ 1 - e^{-\frac{1}{t}} & t > 0 \end{cases}$$

1. Find the median delay and the 90th percentile delay. [5]
2. Derive the probability density function of \(T\). Hence use calculus to find the mean delay. [5]
3. Find the probability that a delay lasts longer than the mean delay. [2]

You are given that the variance of \(T\) is 9.

1. Let \(\overline{T}\) denote the mean of a random sample of 30 delays. Write down an approximation to the distribution of \(\overline{T}\). [3]
2. A random sample of 30 delays is found to have mean 4.2 minutes. Does this cast any doubt on the modelling? [3]

The length of time a battery works, in tens of hours, is modelled by a random variable \(X\) with cumulative distribution function $$F(x) = \begin{cases} 0 & \text{for } x < 0, \\ \frac{x^3}{432}(8-x) & \text{for } 0 \leq x \leq 6, \\ 1 & \text{for } x > 6. \end{cases}$$

1. Find \(P(X > 5)\). [2]
2. A head torch uses three of these batteries. All three batteries must work for the torch to operate. Find the probability that the head torch will operate for more than 50 hours. [2]
3. Show that the upper quartile of the distribution lies between 4·5 and 4·6. [3]
4. Find \(f(x)\), the probability density function for \(X\). [3]
5. Find the mean lifetime of the batteries in hours. [4]
6. The graph of \(f(x)\) is given below. \includegraphics{figure_1} Give a reason why the model may not be appropriate. [1]

It is known that the average lifetime of hair dryers from a certain manufacturer is 2 years. The lifetimes are exponentially distributed.

1. Find the probability that the lifetime of a randomly selected hair dryer is between 1·8 and 2·5 years. [4]
2. Given that 20% of hair dryers have a lifetime of at least \(k\) years, find the value of \(k\). [3]
3. Jon buys his first hair dryer from the manufacturer today. He will replace his hair dryer with another from the same manufacturer immediately when it stops working. Find the probability that, in the next 5 years, Jon will have to replace more than 3 hair dryers. [3]
4. State one assumption that you have made in part (c). [1]

The queueing times, \(T\) minutes, of customers at a local Post Office are modelled by the probability density function $$f(t) = \frac{1}{2500}t(100-t^2) \quad \text{for } 0 \leq t \leq 10,$$ $$f(t) = 0 \quad \text{otherwise.}$$

1. Determine the mean queueing time. [3]
2. 1. Find the cumulative distribution function, \(F(t)\), of \(T\).
   2. Find the probability that a randomly chosen customer queues for more than 5 minutes.
   3. Find the median queueing time. [10]

The random variable \(X\) has probability density function $$f(x) = 1 + \frac{3\lambda x}{2} \quad \text{for } -\frac{1}{2} \leqslant x \leqslant \frac{1}{2},$$ $$f(x) = 0 \quad \text{otherwise,}$$ where \(\lambda\) is an unknown parameter such that \(-1 \leqslant \lambda \leqslant 1\).

1. 1. Find E\((X)\) in terms of \(\lambda\).
   2. Show that \(\text{Var}(X) = \frac{16 - 3\lambda^2}{192}\). [6]
2. Show that P\((X > 0) = \frac{8 + 3\lambda}{16}\). [2]

In order to estimate \(\lambda\), \(n\) independent observations of \(X\) are made. The number of positive observations obtained is denoted by \(Y\) and the sample mean is denoted by \(\overline{X}\).

1. 1. Identify the distribution of \(Y\).
   2. Show that \(T_1\) is an unbiased estimator for \(\lambda\), where $$T_1 = \frac{16Y}{3n} - \frac{8}{3}.$$ [4]
2. 1. Show that \(\text{Var}(T_1) = \frac{64 - 9\lambda^2}{9n}\).
   2. Given that \(T_2\) is also an unbiased estimator for \(\lambda\), where $$T_2 = 8\overline{X},$$ find an expression for Var\((T_2)\) in terms of \(\lambda\) and \(n\).
   3. Hence, giving a reason, determine which is the better estimator, \(T_1\) or \(T_2\). [6]

The probability density function of the continuous random variable \(X\) is given by $$f(x) = \frac{3x^2}{\alpha^3} \quad \text{for } 0 \leq x \leq \alpha$$ $$f(x) = 0 \quad \text{otherwise.}$$ \(\overline{X}\) is the mean of a random sample of \(n\) observations of \(X\).

1. 1. Show that \(U = \frac{4\overline{X}}{3}\) is an unbiased estimator for \(\alpha\). [5]
   2. If \(\alpha\) is an integer, what is the smallest value of \(n\) that gives a rational value for the standard error of \(U\)? [9]
2. \(\overline{X}_1\) and \(\overline{X}_2\) are the means of independent random samples of \(X\), each of size \(n\). The estimator \(V = 4\overline{X}_1 - \frac{8}{3}\overline{X}_2\) is also an unbiased estimator for \(\alpha\).
   1. Show that \(\frac{\text{Var}(U)}{\text{Var}(V)} = \frac{1}{13}\). [4]
   2. Hence state, with a reason, which of \(U\) or \(V\) is the better estimator. [1]

A continuous random variable \(X\) has probability density function \(f\) given by $$f(x) = \begin{cases} \frac{x^2}{a} + b, & 0 \leq x \leq 4 \\ 0 & \text{otherwise} \end{cases}$$ where \(a\) and \(b\) are positive constants. It is given that \(P(X \geq 2) = 0.75\).

1. Show that \(a = 32\) and \(b = \frac{1}{12}\). [5]
2. Find \(E(X)\). [3]
3. Find \(P(X > E(X)|X > 2)\) [4]

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

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

Given that \(E(Y) = 1.75\)

1. Find the values of a and k. [4]
2. Write down the mode of Y [1]

A continuous random variable \(X\) has probability density function given by $$f(x) = \begin{cases} 0.8e^{-0.8x} & x \geq 0, \\ 0 & x < 0. \end{cases}$$

1. Find the mean and variance of \(X\). [4]

The lifetime of a certain organism is thought to have the same distribution as \(X\). The lifetimes in days of a random sample of 60 specimens of the organism were found. The observed frequencies, together with the expected frequencies correct to 3 decimal places, are given in the table.

Range	\(0 \leq x < 1\)	\(1 \leq x < 2\)	\(2 \leq x < 3\)	\(3 \leq x < 4\)	\(x \geq 4\)
Observed	24	22	10	3	1
Expected	33.040	14.846	6.671	2.997	2.446

1. Show how the expected frequency for \(1 \leq x < 2\) is obtained. [4]
2. Carry out a goodness of fit test at the 5\% significance level. [7]

The continuous random variable \(X\) has probability density function given by $$f(x) = \begin{cases} \frac{4}{\pi(1+x^2)} & 0 \leq x \leq 1, \\ 0 & \text{otherwise.} \end{cases}$$

1. Verify that the median value of \(X\) lies between 0.41 and 0.42. [3]
2. Show that E\((X) = \frac{2}{\pi}\ln 2\). [2]
3. Find Var\((X)\). [5]
4. Given that \(\tan\frac{1}{8}\pi = \sqrt{2} - 1\), find the exact value of P(\(X > \frac{1}{4}\sqrt{3}|X > \sqrt{2} - 1\)). [3]