Exponential Distribution

10 The number of hits per minute on a particular website has a Poisson distribution with mean 0.8. The time between successive hits is denoted by \(T\) minutes. Show that \(\mathrm { P } ( T > t ) = \mathrm { e } ^ { - 0.8 t }\) and hence show that \(T\) has a negative exponential distribution. Using a suitable approximation, which should be justified, find the probability that the time interval between the 1st hit and the 51st hit exceeds one hour.

6 The random variable \(X\) has distribution function F given by $$\mathrm { F } ( x ) = \begin{cases} 1 - \mathrm { e } ^ { - 0.6 x } & x \geqslant 0
0 & \text { otherwise } \end{cases}$$ Identify the distribution of \(X\) and state its mean. Find

1. \(\mathrm { P } ( X > 4 )\),
2. the median of \(X\).

7 The waiting time, \(T\) minutes, before a customer is served in a restaurant has distribution function F given by $$\mathrm { F } ( t ) = \begin{cases} 1 - \mathrm { e } ^ { - \lambda t } & t \geqslant 0
0 & t < 0 \end{cases}$$ where \(\lambda\) is a positive constant. The standard deviation of \(T\) is 8 . Find

1. the value of \(\lambda\),
2. the probability that a customer has to wait between 5 and 10 minutes before being served,
3. the median value of \(T\).

7 The random variable \(T\) is the lifetime, in hours, of a randomly chosen decorative light bulb of a particular type. It is given that \(T\) has a negative exponential distribution with mean 1000 hours.

1. Write down the probability density function of \(T\).
2. Find the probability that a randomly chosen bulb of this type has a lifetime of more than 2000 hours. A display uses 10 randomly chosen bulbs of this type, and they are all switched on simultaneously. Find the greatest value of \(t\) such that the probability that they are all alight at time \(t\) hours is at least 0.9 .

7 The random variable \(X\) has probability density function f given by $$\mathrm { f } ( x ) = \begin{cases} 0.2 \mathrm { e } ^ { - 0.2 x } & x \geqslant 0
0 & \text { otherwise } \end{cases}$$

1. Find the distribution function of \(X\).
2. Find \(\mathrm { P } ( X > 2 )\).
3. Find the median of \(X\).

8 A certain mechanical component has a lifetime, \(T\) months, which has a negative exponential distribution with mean 2.5.

1. A machine is fitted with 5 of these components which function independently.
   1. Find the probability that all 5 components are operating 3 months after being fitted.
   2. Find also the probability that exactly two components fail within one month of being fitted.
2. Show that the probability that \(n\) independent components are all operating \(c\) months after being fitted is equal to the probability that a single component is operating \(n c\) months after being fitted.

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. $$\mathrm { F } ( t ) = \begin{cases} 0 & t \leqslant 0
1 - \mathrm { e } ^ { - \frac { 1 } { 3 } t } & t > 0 \end{cases}$$

1. Find the median delay and the 90th percentile delay.
2. Derive the probability density function of \(T\). Hence use calculus to find the mean delay.
3. Find the probability that a delay lasts longer than the mean delay. You are given that the variance of \(T\) is 9 .
4. Let \(\bar { T }\) denote the mean of a random sample of 30 delays. Write down an approximation to the distribution of \(\bar { T }\).
5. A random sample of 30 delays is found to have mean 4.2 minutes. Does this cast any doubt on the modelling?

9 The continuous random variable \(X\) has probability density function given by $$\mathrm { f } ( x ) = \begin{cases} 0 & x < 2
a \mathrm { e } ^ { - ( x - 2 ) } & x \geqslant 2 \end{cases}$$ where \(a\) is a constant. Show that \(a = 1\). Find the distribution function of \(X\) and hence find the median value of \(X\). The random variable \(Y\) is defined by \(Y = \mathrm { e } ^ { X }\). Find

1. the probability density function of \(Y\),
2. \(\mathrm { P } ( Y > 10 )\).

7 The random variable \(T\) is the lifetime, in hours, of a particular type of battery. It is given that \(T\) has a negative exponential distribution with mean 500 hours.

1. Write down the probability density function of \(T\).
2. Find the probability that a randomly chosen battery of this type has a lifetime of more than 750 hours.
3. Find the median value of \(T\).

7 The lifetime, \(x\) years, of the power light on a freezer, which is left on continuously, can be modelled by the continuous random variable with density function given by $$\mathrm { f } ( x ) = \begin{cases} k \mathrm { e } ^ { - 3 x } & x > 0
0 & \text { otherwise } \end{cases}$$ where \(k\) is a constant.

1. Show that \(k = 3\).
2. Find the lower quartile.
3. Find the mean lifetime.

7 The random variable \(X\) has an exponential distribution with parameter \(\lambda\) 7

1. Prove that \(\mathrm { E } ( X ) = \frac { 1 } { \lambda }\)
   7
2. Prove that \(\operatorname { Var } ( X ) = \frac { 1 } { \lambda ^ { 2 } }\)

8 The lifetime, in years, of an electrical component is the random variable \(T\), with probability density function f given by $$\mathrm { f } ( t ) = \begin{cases} A \mathrm { e } ^ { - \lambda t } & t \geqslant 0
0 & \text { otherwise } \end{cases}$$ where \(A\) and \(\lambda\) are positive constants.

1. Show that \(A = \lambda\). It is known that out of 100 randomly chosen components, 16 failed within the first year.
2. Find an estimate for the value of \(\lambda\), and hence find an estimate for the median value of \(T\).

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

1. Verify that \(\int _ { 0 } ^ { \infty } \mathrm { f } ( x ) \mathrm { d } x = 1\) and sketch \(\mathrm { f } ( x )\).
2. In this part of the question you may use the following result. $$\int _ { 0 } ^ { \infty } x ^ { r } \mathrm { e } ^ { - \lambda x } \mathrm {~d} x = \frac { r ! } { \lambda ^ { r + 1 } } \quad \text { for } r = 0,1,2 , \ldots$$ Derive the mean and variance of \(X\) in terms of \(\lambda\). 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.
3. Let \(\bar { X }\) denote the mean lifetime, in years, of a random sample of 50 appliances. Write down an approximate distribution for \(\bar { X }\).
4. 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?

8 The time in hours to failure of a component may be modelled by an exponential distribution with parameter \(\lambda = 0.025\) In a manufacturing process, the machine involved uses one of these components continuously until it fails. The component is then immediately replaced.
8

1. Write down the mean time to failure for a component. 8
2. Find the probability that a component will fail during a 12-hour shift. 8
3. A component has not failed for 30 hours. Find the probability that this component lasts for at least another 30 hours.
   [0pt] [2 marks] 8
4. Find the probability that a component does not fail during 4 consecutive 12-hour shifts.
   [0pt] [3 marks]
   8
5. 1. State the distribution that can be used to model the number of components that fail during one hour of the manufacturing process.
      [0pt] [2 marks]
      8
6. (ii) Hence, or otherwise, find the probability that no components fail during 5 consecutive 12-hour shifts.
   [0pt] [2 marks]

5 Layla works at an internet café. Each terminal at the café has its own keyboard, and keyboards need to be replaced whenever faults develop. Layla knows that the number of weeks for which a keyboard lasts before it needs to be replaced can be modelled by the random variable \(X\), which has an exponential distribution with mean 20 and variance 400 . She wants to investigate how likely it is that the keyboard at a terminal will need to be replaced at least 3 times within a year (taken as being a period of 52 weeks). Layla designs the simulation shown in the spreadsheet below. Each of the 20 rows below the heading row consists of 3 values of \(X\) together with their sum \(T\). All of the values in the spreadsheet have been rounded to 1 decimal place.

1. Explain why \(T\) represents the number of weeks after which the third keyboard at a terminal will need to be replaced.
2. Use the information in the spreadsheet to write down an estimate of \(\mathrm { P } ( T > 52 )\).
3. Explain how you could obtain a more reliable estimate of \(\mathrm { P } ( T > 52 )\).
4. The internet café has 50 terminals. You are given that faults in keyboards occur independently of each other. Determine an estimate of the probability that the mean number of weeks before which the third keyboard at a terminal needs to be replaced is more than 52 .

5. The continuous random variable \(T\) has cumulative distribution function $$\mathrm { F } ( t ) = \begin{cases} 0 & t < 0 ,
1 - \mathrm { e } ^ { - 0.25 t } & t \geqslant 0 . \end{cases}$$

1. Find the cumulative distribution function of \(2 T\).
2. Show that, for constant \(k , \quad \mathrm { E } \left( \mathrm { e } ^ { k T } \right) = \frac { 1 } { 1 - 4 k }\). You should state with a reason the range of values of \(k\) for which this result is valid.
3. \(T\) is the time before a certain event occurs. Show that the probability that no event occurs between time \(T = 0\) and time \(T = \theta\) is the same as the probability that the value of a random variable with the distribution \(\operatorname { Po } ( \lambda )\) is 0 , for a certain value of \(\lambda\). You should state this value of \(\lambda\) in terms of \(\theta\).
   [0pt] [BLANK PAGE]
   [0pt] [BLANK PAGE]
   [0pt] [BLANK PAGE]
   [0pt] [BLANK PAGE]
   [0pt] [BLANK PAGE]

4
8
16
256 2 The random variable \(T\) has an exponential distribution with mean 2 Find \(\mathrm { P } ( T \leq 1.4 )\)
Circle your answer.
\(\mathrm { e } ^ { - 2.8 }\)
\(\mathrm { e } ^ { - 0.7 }\)
\(1 - e ^ { - 0.7 }\)
\(1 - \mathrm { e } ^ { - 2.8 }\) The continuous random variable \(Y\) has cumulative distribution function $$\mathrm { F } ( y ) = \left\{ \begin{array} { l r } 0 & y < 2
- \frac { 1 } { 9 } y ^ { 2 } + \frac { 10 } { 9 } y - \frac { 16 } { 9 } & 2 \leq y < 5
1 & y \geq 5 \end{array} \right.$$ Find the median of \(Y\) Circle your answer. 2
\(\frac { 10 - 3 \sqrt { 2 } } { 2 }\)
\(\frac { 7 } { 2 }\)
\(\frac { 10 + 3 \sqrt { 2 } } { 2 }\) Turn over for the next question 4 Research has shown that the mean number of volcanic eruptions on Earth each day is 20 Sandra records 162 volcanic eruptions during a period of one week. Sandra claims that there has been an increase in the mean number of volcanic eruptions per week. Test Sandra's claim at the \(5 \%\) level of significance.