ALTA - CHAPTER 5 - CONTINUOUS RANDOM VARIABLES Questions and Correct Answers the Latest Update
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ALTA
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ALTA
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?
0.500
Let X be the...
ALTA - CHAPTER 5 - CONTINUOUS
RANDOM VARIABLES Questions and
Correct Answers the Latest Update
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?
✓ 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=1/8
✓ 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
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
✓ 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
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?
✓ $\text{std=}2.0\text{min}$std=2.0min
✓ The standard deviation for the waiting time is
✓ σσ=(b−a)212−−−−−−−√=(b−a)12−−√
✓ In this case we find that the standard deviation isσ=11−412−−√=712−−√=73-
√6≈2.0So the standard deviation for waiting is approximately 2.0 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
average (mean) time for how long she will wait?
✓ $\text{mean=}7.5\text{min}$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.
If X∼U(5.5,18.5) is a continuous uniform random variable, what is P(X<9)?
✓ 7/26
✓ We are interested in values less than 9. The probability is the area of the rectangle
under this part of the curve. Area is the product of the base (length of this interval)
and the height, which is the value of the probability density function(1b−a).So we
findP(X<9)=(9−5.5)(1/18.5−5.5)=(3.5)(1/13)=7/26
The amount of time it takes Victoria to solve crossword puzzles can be modeled
by a continuous and uniformly distributed random time between 4 minutes and
12 minutes. What is the probability that it will take Victoria greater than 8
minutes (total) for a puzzle that she has already been working on for 7 minutes?
Provide the final answer as a fraction
✓ $\frac{4}{5}$45
✓ Let X be the time it takes to solve a crossword pule. We are told that the
probability density function for X is uniform, so we can conclude that the conditional
probability density function for X given that X>7 is also a uniform distribution. The
possible values are restricted to the range 7 to 12. This interval will help us write the
conditional probability density function (height of the rectangle).
f(x)=112−7=15
We are interested in values greater than 8. So the range of desired outcomes is
8 to 12. This will be the length of the 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
findP(X>8∣∣X>7)=(12−8)(112−7)=(4)(15)=45zz
The amount of time it takes Ariana to do a math problem is continuous and
uniformly distributed between 38 seconds and 79 seconds. What is the
probability that it takes Ariana between 58 and 70 seconds to do a math
✓ 0.293
✓ Let X be the time it takes to do a math problem. The range of possible values is 38
to 79, so X∼U(38,79).We are interested in values greater than 58 and less than 70.
The probability is the area of the rectangle under this part of the curve. Area is the
product of the base (length of this interval) and the height, which is the value of the
probability density function(1b−a).So we
findP(58<X<70)=(70−58)(179−38)=(12)(141)=0.293
The probability density function for the random variable X is shown below. Find
P(X<6).
✓ 6/7
✓ Remember that the probability for a range of values is equal to the area under the
curve. So we want the area under the curve for the values of x less than 6.
✓ A coordinate system has a horizontal axis labeled from 0 to 9 in increments of 1 and
a vertical axis labeled from 0 to StartFraction 1 Over 7 EndFraction. A solid line
segment labeled f left-parenthesis x right-parenthesis starts at the point left
parenthesis 0 comma StartFraction 1 Over 7 EndFraction right-parenthesis and
extends horizontally to the right until it reaches the point left-parenthesis 6 comma
StartFraction 1 Over 7 EndFraction right-parenthesis. The line segment then
decreases from left to right connecting the points left-parenthesis 6 comma
StartFraction 1 Over 7 EndFraction right-parenthesis and left-parenthesis 8 comma
StartFraction 1 Over 7 EndFraction right-parenthesis. A dashed vertical line segment
connects the points left-parenthesis 6 comma 0 right-parenthesis and left-parenthesis
6 comma StartFraction 1 Over 7 EndFraction right-parenthesis. A rectangular region
to the left of the vertical line segment and underneath the solid line segment is
shaded.
✓ Note that this region is a rectangle, with base 6 and height 17, so the area (and
hence the probability) isP(X<6)=(base)(height)=(6)(17)=67So we find that
P(X<6)=67.
The probability density function for the random variable X is shown below. Find
P(X>4).
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