5.03c Calculate mean/variance: by integration

\(X\) is a continuous random variable with the distribution N\((48.5, 12.5^2)\). The values of \(X\) are transformed to standardised values of \(Y\), using the equation \(Y = aX + b\), where \(a\) and \(b\) are constants with \(a > 0\).

1. Find values of \(a\) and \(b\) for which the mean and standard deviation of \(Y\) are 40 and 10 respectively. [4]
2. State the distribution of \(Y\). [1]

Two randomly chosen standardised values are denoted by \(Y_1\) and \(Y_2\).

1. Calculate the probability that \(Y_2\) is at least 10 greater than \(Y_1\). [5]

The continuous random variable \(Y\) has probability density function given by $$\text{f}(y) = \begin{cases} -\frac{1}{4}y & -2 < y < 0, \\ \frac{1}{4}y & 0 \leqslant y \leqslant 2, \\ 0 & \text{otherwise.} \end{cases}$$ Find

1. the interquartile range of \(Y\), [4]
2. Var\((Y)\), [5]
3. E\((|Y|)\). [4]

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]

Geoffrey is a university lecturer. He has to prepare five questions for an examination. He knows by experience that it takes about 3 hours to prepare a question, and he models the time (in minutes) taken to prepare one by the Normally distributed random variable \(X\) with mean 180 and standard deviation 12, independently for all questions.

1. One morning, Geoffrey has a gap of 2 hours 50 minutes (170 minutes) between other activities. Find the probability that he can prepare a question in this time. [3]
2. One weekend, Geoffrey can devote 14 hours to preparing the complete examination paper. Find the probability that he can prepare all five questions in this time. [3]

A colleague, Helen, has to check the questions.

1. She models the time (in minutes) to check a question by the Normally distributed random variable \(Y\) with mean 50 and standard deviation 6, independently for all questions and independently of \(X\). Find the probability that the total time for Geoffrey to prepare a question and Helen to check it exceeds 4 hours. [3]
2. When working under pressure of deadlines, Helen models the time to check a question in a different way. She uses the Normally distributed random variable \(\frac{1}{2}X\), where \(X\) is as above. Find the length of time, as given by this model, which Helen needs to ensure that, with probability 0.9, she has time to check a question. [4]

Ian, an educational researcher, suggests that a better model for the time taken to prepare a question would be a constant \(k\) representing "thinking time" plus a random variable \(T\) representing the time required to write the question itself, independently for all questions.

1. Taking \(k\) as 45 and \(T\) as Normally distributed with mean 120 and standard deviation 10 (all units are minutes), find the probability according to Ian's model that a question can be prepared in less than 2 hours 30 minutes. [2]

Juliet, an administrator, proposes that the examination should be reduced in time and shorter questions should be used.

1. Juliet suggests that Ian's model should be used for the time taken to prepare such shorter questions but with \(k = 30\) and \(T\) replaced by \(\frac{2}{3}T\). Find the probability as given by this model that a question can be prepared in less than \(1\frac{1}{4}\) hours. [3]

A random variable \(X\) has an exponential distribution with probability density function \(f(x) = \lambda e^{-\lambda x}\) for \(x \geq 0\), where \(\lambda\) is a positive constant.

1. Verify that \(\int_0^{\infty} f(x) \, dx = 1\) and sketch \(f(x)\). [5]
2. In this part of the question you may use the following result. $$\int_0^{\infty} x^r e^{-\lambda x} \, dx = \frac{r!}{\lambda^{r+1}} \text{ for } r = 0, 1, 2, \ldots$$ Derive the mean and variance of \(X\) in terms of \(\lambda\). [6]

The random variable \(X\) is used to model the lifetime, in years, of a particular type of domestic appliance. The manufacturer of the appliance states that, based on past experience, the mean lifetime is 6 years.

1. Let \(\overline{X}\) denote the mean lifetime, in years, of a random sample of 50 appliances. Write down an approximate distribution for \(\overline{X}\). [4]
2. A random sample of 50 appliances is found to have a mean lifetime of 7.8 years. Does this cast any doubt on the model? [3]

\includegraphics{figure_6} Figure 1 shows a square of side 1 and area \(l^2\) which lies in the first quadrant with one vertex at the origin. A point \(P\) with coordinates \((X, Y)\) is selected at random inside the square and the coordinates are used to estimate \(l^2\). It is assumed that \(X\) and \(Y\) are independent random variables each having a continuous uniform distribution over the interval \([0, l]\). [You may assume that E\((X^n Y^m) = \) E\((X^n)\)E\((Y^m)\), where \(n\) is a positive integer.]

1. Use integration to show that E\((X^n) = \frac{l^{n+1}}{n+1}\). [3]

The random variable \(S = kXY\), where \(k\) is a constant, is an unbiased estimator for \(l^2\).

1. [(b)] Find the value of \(k\). [3]
2. Show that Var \(S = \frac{7l^4}{9}\). [3]

The random variable \(U = q(X^2 + Y^2)\), where \(q\) is a constant, is also an unbiased estimator for \(l^2\).

1. [(d)] Show that the value of \(q = \frac{3}{2}\). [3]
2. Find Var \(U\). [3]
3. State, giving a reason, which of \(S\) and \(U\) is the better estimator of \(l^2\). [1]

The point (2, 3) is selected from inside the square.

1. [(g)] Use the estimator chosen in part (f) to find an estimate for the area of the square. [1]

TOTAL FOR PAPER: 75 MARKS

The continuous random variable \(X\) has probability density function $$f(x) = \begin{cases} \frac{4}{45}(x^3 - 10x^2 + 29x - 20) & 1 \leq x \leq 4 \\ 0 & \text{otherwise} \end{cases}$$

1. Find P\((X < 2)\) [2 marks]
2. Verify that the median of \(X\) is 2.3, correct to two significant figures. [4 marks]
3. Find the mean of \(X\). [2 marks]

The continuous random variable \(X\) has probability density function $$f(x) = \begin{cases} \frac{k}{x^n} & x \geqslant 1, \\ 0 & \text{otherwise}, \end{cases}$$ where \(n\) and \(k\) are constants and \(n\) is an integer greater than 1.

1. Find \(k\) in terms of \(n\). [3]
2. 1. When \(n = 4\), find the cumulative distribution function of \(X\). [3]
   2. Hence determine P\((X > 7 | X > 5)\) when \(n = 4\). [4]
3. Determine the values of \(n\) for which Var\((X)\) is not defined. [5]

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]

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]

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 continuous random variable \(X\) has cumulative distribution function given by $$F(x) = \begin{cases} 0 & x \leq 0 \\ k\left(x^3 - \frac{3}{8}x^4\right) & 0 < x \leq 2 \\ 1 & x > 2 \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = \frac{1}{2}\) [1]
2. Showing your working clearly, use calculus to find
   1. E(\(X\))
   2. the mode of \(X\)
   [6]

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 $$f(x) = \begin{cases} 3e^{-x} & 0 \leq x \leq k, \\ 0 & \text{otherwise,} \end{cases}$$ where \(k\) is a constant.

1. Show that \(e^{-k} = \frac{2}{3}\). [2]
2. Show that the moment generating function of \(X\) is given by \(M_X(t) = \frac{3}{1-t}\left(1 - \frac{2}{3}e^{kt}\right)\). [4]
3. By expanding \(M_X(t)\) as a power series in \(t\), up to and including the term in \(t^2\), show that $$M_X(t) = 1 + (1 - 2k)t + (1 - 2k - k^2)t^2 + \ldots.$$ [3] [You may use the standard series for \((1-t)^{-1}\) and \(e^{kt}\) without proof.]
4. Deduce that the exact value of E\((X)\) is \(1 - 2\ln\left(\frac{2}{3}\right)\). [1]

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]