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 questions · Moderate -0.1

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CAIE FP2 2019 June Q6
6 marks Moderate -0.8
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.
CAIE FP2 2014 November Q7
6 marks Standard +0.3
The time, \(T\) seconds, between successive cars passing a particular checkpoint on a wide road has probability density function f given by $$f(t) = \begin{cases} \frac{1}{100}e^{-0.01t} & t \geq 0, \\ 0 & \text{otherwise.} \end{cases}$$
  1. State the expected value of \(T\). [1]
  2. Find the median value of \(T\). [3]
Sally wishes to cross the road at this checkpoint and she needs 20 seconds to complete the crossing. She decides to start out immediately after a car passes. Find the probability that she will complete the crossing before the next car passes. [2]
CAIE S2 2020 Specimen Q4
10 marks Standard +0.3
The lifetimes, in hours, of light bulbs have an exponential distribution with parameter \(\frac{1}{500}\). Each bulb is tested and rejected if the lifetime is less than 500 hours.
  1. Find the probability that a bulb of this type has a lifetime of more than 500 hours. [4]
  2. Find the probability that the lifetime is at least three times the expected lifetime. [6]