Exponential Distribution

39 questions · 18 question types identified

Sort by: Question count | Difficulty
Link Poisson to exponential

A question is this type if and only if it derives the exponential distribution of waiting times from a Poisson process with a given rate.

5 Standard +0.6
12.8% of questions
5.02i Poisson distribution: random events model5.03a Continuous random variables: pdf and cdf
Show example »
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.
View full question →
State distribution and mean

A question is this type if and only if it asks to identify or state the type of distribution and its mean from a given PDF or CDF.

5 Moderate -0.7
12.8% of questions
5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf
Show example »
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 \(X\).
View full question →
Derive CDF from PDF

A question is this type if and only if it asks to find the cumulative distribution function by integrating a given probability density function.

4 Moderate -0.1
10.3% of questions
5.03a Continuous random variables: pdf and cdf5.03e Find cdf: by integration
Show example »
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 \(X\).
View full question →
Find threshold for given probability

A question is this type if and only if it asks to find a value of n, t, or d such that a probability condition is satisfied.

4 Standard +0.1
10.3% of questions
5.03a Continuous random variables: pdf and cdf
Show example »
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 .
View full question →
Calculate mean or variance

A question is this type if and only if it asks to find, state, or prove the mean or variance of an exponential distribution, either from the PDF or using calculus.

3 Standard +0.3
7.7% of questions
Show example »
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 \(\operatorname { Var } ( X ) = \frac { 1 } { \lambda ^ { 2 } }\)
View full question →
Multiple independent components

A question is this type if and only if it involves calculating probabilities for multiple independent components or items, each with exponential lifetime.

3 Standard +0.5
7.7% of questions
Show example »
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.
View full question →
Calculate probability with given parameter

Question provides the complete PDF or CDF with a specific numerical parameter value and asks to calculate probabilities or percentiles directly.

3 Moderate -0.1
7.7% of questions
Show example »
6 The random variable \(T\) is the lifetime, in hours, of a randomly chosen battery of a particular type. It is given that \(T\) has a negative exponential distribution with mean 400 hours.
  1. Write down the probability density function of \(T\).
  2. Find the probability that a battery of this type has a lifetime that is less than 500 hours.
  3. Find the median of the distribution.
View full question →
Find constant in PDF

A question is this type if and only if it asks to find or show the value of a constant in a probability density function by using the property that the integral equals 1.

2 Standard +0.0
5.1% of questions
Show example »
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.
View full question →
Transform exponential random variable

A question is this type if and only if it involves finding the distribution or PDF of a transformed variable Y = g(X) where X is exponential.

2 Standard +0.8
5.1% of questions
5.03a Continuous random variables: pdf and cdf5.03e Find cdf: by integration
Show example »
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. \(\mathrm { P } ( Y > 10 )\).
View full question →
Normal approximation to sum

A question is this type if and only if it uses a normal approximation to find probabilities for the sum of multiple exponential random variables.

2 Standard +0.3
5.1% of questions
Show example »
In a certain fluid, bacteria are distributed randomly and occur at a constant average rate of 2.5 in every 10 ml of the fluid.
  1. State a further condition needed for the number of bacteria in a fixed volume of the fluid to be well modelled by a Poisson distribution, explaining what your answer means. [2]
Assume now that a Poisson model is appropriate.
  1. Find the probability that in 10 ml there are at least 5 bacteria. [2]
  2. Find the probability that in 3.7 ml there are exactly 2 bacteria. [3]
  3. Use a suitable approximation to find the probability that in 1000 ml there are fewer than 240 bacteria, justifying your approximation. [7]
View full question →
Find parameter from given information

Question asks to find the parameter λ (or related constant) from given information such as mean, standard deviation, or median before calculating probabilities.

2 Standard +0.3
5.1% of questions
5.03a Continuous random variables: pdf and cdf5.03b Solve problems: using pdf
Show example »
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 \(T\).
View full question →
Find quartiles or percentiles

A question is this type if and only if it asks to find specific percentiles, quartiles, or median values of an exponential distribution using the cumulative distribution function.

1 Standard +0.3
2.6% of questions
5.03f Relate pdf-cdf: medians and percentiles
Show example »
3 The time, in hours, until an electronic component fails is represented by the random variable \(X\). In this question two models for \(X\) are proposed.
  1. In one model, \(X\) has cumulative distribution function $$\mathrm { G } ( x ) = \begin{cases} 0 & x \leqslant 0 \\ 1 - \left( 1 + \frac { x } { 200 } \right) ^ { - 2 } & x > 0 \end{cases}$$ (A) Sketch \(\mathrm { G } ( x )\).
    (B) Find the interquartile range for this model. Hence show that a lifetime of more than 454 hours (to the nearest hour) would be classed as an outlier.
  2. In the alternative model, \(X\) has probability density function $$\mathrm { f } ( x ) = \begin{cases} \frac { 1 } { 200 } \mathrm { e } ^ { - \frac { 1 } { 200 } x } & x > 0 \\ 0 & \text { elsewhere. } \end{cases}$$ (A) For this model show that the cumulative distribution function of \(X\) is $$\mathrm { F } ( x ) = \begin{cases} 0 & x \leqslant 0 \\ 1 - \mathrm { e } ^ { - \frac { 1 } { 200 } x } & x > 0 \end{cases}$$ (B) Show that \(\mathrm { P } ( X > 50 ) = \mathrm { e } ^ { - 0.25 }\).
    (C) It is observed that a particular component is still working after 400 hours. Find the conditional probability that it will still be working after a further 50 hours (i.e. after a total of 450 hours) given that it is still working after 400 hours.
View full question →
Conditional probability or memoryless

A question is this type if and only if it asks to find conditional probabilities given that a component has already lasted a certain time, using the memoryless property.

1 Standard +0.3
2.6% of questions
Show example »
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. (i) 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 (e) (ii) Hence, or otherwise, find the probability that no components fail during 5 consecutive 12-hour shifts.
    [0pt] [2 marks]
View full question →
Multiple choice identification

A question is this type if and only if it presents a multiple choice question asking to identify a probability expression or value for an exponential distribution.

1 Moderate -0.3
2.6% of questions
Show example »
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.
View full question →
Derive PDF from CDF

A question is this type if and only if it asks to find the probability density function by differentiating a given cumulative distribution function.

0
0.0% of questions
Estimate parameter from data

A question is this type if and only if it asks to estimate the parameter λ or mean from given sample data or failure rates.

0
0.0% of questions
Verify exponential distribution properties

A question is this type if and only if it asks to verify or show that a given function satisfies properties of an exponential distribution (e.g., integral equals 1, correct form).

0
0.0% of questions
Expectation of function of X

A question is this type if and only if it asks to find E(g(X)) for some function g, such as E(e^(kT)), using integration.

0
0.0% of questions
Show example »
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]
    [0pt]
    [0pt]
    [0pt]
    [0pt]
View full question →
Unclassified

Questions not yet assigned to a type.

1
2.6% of questions
Show 1 unclassified »
Answer only one of the following two alternatives. **EITHER** A particle of mass 0.1 kg lies on a smooth horizontal table on the line between two points \(A\) and \(B\) on the table, which are 6 m apart. The particle is joined to \(A\) by a light elastic string of natural length 2 m and modulus of elasticity 60 N, and to \(B\) by a light elastic string of natural length 1 m and modulus of elasticity 20 N. The mid-point of \(AB\) is \(M\), and \(O\) is the point between \(M\) and \(B\) at which the particle can rest in equilibrium. Show that \(MO = 0.2\) m. [4] The particle is held at \(M\) and then released. Show that the equation of motion is $$\frac{\mathrm{d}^2y}{\mathrm{d}t^2} = -500y,$$ where \(y\) metres is the displacement from \(O\) in the direction \(OB\) at time \(t\) seconds, and state the period of the motion. [5] For the instant when the particle is 0.3 m from \(M\) for the first time, find
  1. the speed of the particle, [2]
  2. the time taken, after release, to reach this position. [3]
**OR** The continuous random variable \(T\) has a negative exponential distribution with probability density function given by $$\mathrm{f}(t) = \begin{cases} \lambda\mathrm{e}^{-\lambda t} & t \geqslant 0, \\ 0 & \text{otherwise.} \end{cases}$$ Show that for \(t \geqslant 0\) the distribution function is given by F\((t) = 1 - \mathrm{e}^{-\lambda t}\). [2] The table below shows some values of F\((t)\) for the case when the mean is 20. Find the missing value. [2]
\(t\)0510152025303540
F\((t)\)00.22120.39350.63210.71350.77690.82620.8647
It is thought that the lifetime of a species of insect under laboratory conditions has a negative exponential distribution with mean 20 hours. When observation starts there are 100 insects, which have been randomly selected. The lifetimes of the insects, in hours, are summarised in the table below.
Lifetime (hours)\(0-5\)\(5-10\)\(10-15\)\(15-20\)\(20-25\)\(25-30\)\(30-35\)\(35-40\)\(\geqslant 40\)
Frequency2020119985117
Calculate the expected values for each interval, assuming a negative exponential model with a mean of 20 hours, giving your values correct to 2 decimal places. [3] Perform a \(\chi^2\)-test of goodness of fit, at the 5% level of significance, in order to test whether a negative exponential distribution, with a mean of 20 hours, is a suitable model for the lifetime of this species of insect under laboratory conditions. [7]