Knewton Alta - Chapter 5 - Continuous
Random Variables 2022
The amount of time it takes Hugo to solve a Rubik's cube is continuous and
uniformly distributed between 4 minutes and 11 minutes. What is the probability
...
Knewton Alta - Chapter 5 - Continuous
Random Variables 2022
The amount of time it takes Hugo to solve a Rubik's cube is continuous and
uniformly distributed between 4 minutes and 11 minutes. What is the probability that
it takes Hugo less than 5 minutes given that it takes less than 6 minutes for him to
solve a Rubik's cube?
Provide the final answer as a fraction. -
Uniform distribution - happens when each of the values within an interval are equally
likely to occur, so each value has the exact same probability as the others over the
entire interval givenA Uniform distribution may also be referred to as a Rectangular
distribution
Conditional probability - the likelihood that an event will occur given that another
event has already occurred
The amount of time it takes Evelyn to solve a Rubik's cube is continuous and
uniformly distributed between 3 minutes and 12 minutes. What is the probability that
it takes Evelyn more than 8 minutes given that it takes more than 4 minutes for her to
solve a Rubik's cube? - Correct answer:
0.500
Let X be the time it takes to solve a Rubik's cube. The range of possible values is 3
to 12, so X∼U(3,12).
We are given that it takes more than 4 minutes, so this means that the possible
values are restricted to the range 4 to 12. This interval will help us write our
probability density function (height of the rectangle).
f(x)=1//12−4=18
We are interested in values greater than 8. So the range of desired outcomes is 8 to
12. This will be the length of our base, 12−8=4.
The probability is the area of the shaded rectangle under the probability density
function line, which is the product of the base and the height. So we find
P(X>8∣∣X>4)=(12−8)(112−4)=(4)(18)=48=0.500
Lexie is waiting for a train. If X is the amount of time before the next train arrives, and
X is uniform with values between 4 and 11 minutes, then what is the average (mean)
time for how long she will wait? - Correct
answers:$\text{mean=}7.5\text{min}$mean=7.5min
The mean of a random variable with a uniform distribution U(a,b) is
μ=a+b2
(this is the midpoint of the interval [a,b]). So in this case, a=4 and b=11, so the mean
isa+b2=4+112=152=7.5So the average waiting time is 7.5 minutes.
Lexie is waiting for a train. If X is the amount of time before the next train arrives, and
X is uniform with values between 4 and 11 minutes, then what is the approximate
standard deviation for how long she will wait, rounded to one decimal place? -
Correct answers:$\text{std=}2.0\text{min}$std=2.0min...
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