5.03a Continuous random variables: pdf and cdf

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]

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]

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]

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]

A continuous random variable \(X\) has cumulative distribution function \(F\) given by $$F(x) = \begin{cases} 0 & \text{for } x < 0, \\ \frac{1}{4}x & \text{for } 0 \leqslant x \leqslant 2, \\ \frac{1}{480}x^4 + \frac{7}{15} & \text{for } 2 < x \leqslant b, \\ 1 & \text{for } x > b. \end{cases}$$

1. Show that \(b = 4\). [2]
2. Find P\((X \leqslant 2 \cdot 5)\). [2]
3. Write down the value of the lower quartile of \(X\). [1]
4. Find the value of the upper quartile of \(X\). [3]
5. Find, correct to three significant figures, the value of \(k\) that satisfies the equation P\((X > 3 \cdot 5) = \text{P}(X < k)\). [4]

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]

Customers arrive at a shop such that the number of arrivals in a time interval of \(t\) minutes follows a Poisson distribution with mean \(0.5t\).

1. Find the probability that exactly 5 customers arrive between 11 a.m. and 11.15 a.m. [3]
2. A customer arrives at exactly 11 a.m.
   1. Let the next customer arrive at \(T\) minutes past 11 a.m. Show that $$P(T > t) = e^{-0.5t}.$$
   2. Hence find the probability density function, \(f(t)\), of \(T\).
   3. Hence, giving a reason, write down the mean and the standard deviation of the time between the arrivals of successive customers. [7]

The continuous random variable \(X\) is uniformly distributed over the interval \((\theta - 1, \theta + 5)\), where \(\theta\) is an unknown constant.

1. Find the mean and the variance of \(X\). [2]
2. Let \(\overline{X}\) denote the mean of a random sample of 9 observations of \(X\). Find, in terms of \(\overline{X}\), an unbiased estimator for \(\theta\) and determine its standard error. [4]

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]

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]

The length of time, in years, that a salesman keeps his company car may be modelled by the continuous random variable \(T\). The probability density function of \(T\) is given by $$f(t) = \begin{cases} \frac{2}{5}t e^{-\frac{t^2}{5}} & t \geq 0, \\ 0 & \text{otherwise}. \end{cases}$$

1. Sketch the graph of \(f(t)\). [2]
2. Find the cumulative distribution function \(F(t)\) and hence find the median value of \(T\). [3]
3. Find the probability that \(T\) is greater than the modal value of \(T\). [5]
4. The probability that a randomly chosen salesman keeps his car longer than \(N\) years is \(0.05\). Find the value of \(N\) correct to \(3\) significant figures. [3]