Maximum or minimum of uniforms

Questions asking about the distribution or probabilities of the maximum or minimum of several independent uniform variables.

3 questions

Edexcel S2 2009 June Q6
6. The three independent random variables \(A , B\) and \(C\) each has a continuous uniform distribution over the interval \([ 0,5 ]\).
  1. Find \(\mathrm { P } ( A > 3 )\).
  2. Find the probability that \(A , B\) and \(C\) are all greater than 3 . The random variable \(Y\) represents the maximum value of \(A , B\) and \(C\). The cumulative distribution function of \(Y\) is $$\mathrm { F } ( y ) = \begin{cases} 0 & y < 0
    \frac { y ^ { 3 } } { 125 } & 0 \leqslant y \leqslant 5
    1 & y > 5 \end{cases}$$
  3. Find the probability density function of \(Y\).
  4. Sketch the probability density function of \(Y\).
  5. Write down the mode of \(Y\).
  6. Find \(\mathrm { E } ( Y )\).
  7. Find \(\mathrm { P } ( Y > 3 )\).
Edexcel S2 2011 June Q4
  1. In a game, players select sticks at random from a box containing a large number of sticks of different lengths. The length, in cm , of a randomly chosen stick has a continuous uniform distribution over the interval [7,10].
A stick is selected at random from the box.
  1. Find the probability that the stick is shorter than 9.5 cm . To win a bag of sweets, a player must select 3 sticks and wins if the length of the longest stick is more than 9.5 cm .
  2. Find the probability of winning a bag of sweets. To win a soft toy, a player must select 6 sticks and wins the toy if more than four of the sticks are shorter than 7.6 cm .
  3. Find the probability of winning a soft toy.
Edexcel FS2 2024 June Q4
  1. The random variable \(G\) has a continuous uniform distribution over the interval \([ - 3,15 ]\)
    1. Calculate \(\mathrm { P } ( G > 12 )\)
    The random variable \(H\) has a continuous uniform distribution over the interval [2, w] The random variables \(G\) and \(H\) are independent and \(\mathrm { E } ( H ) = 10\)
  2. Show that the probability that \(G\) and \(H\) are both greater than 12 is \(\frac { 1 } { 16 }\) The random variable \(A\) is the area on a coordinate grid bounded by $$\begin{aligned} & y = - 3
    & y = - 4 | x | + k \end{aligned}$$ where \(k\) is a value from the continuous uniform distribution over the interval [5,10]
  3. Calculate the expected value of \(A\)