Random Variables - AP Statistics
Card 0 of 20
Variable 
has a Poisson distribution with a mean of 
. What is the variance of Variable 
?
Variable
Because 
has a Poisson distribution we know that:
and
.
Therefore, since we are given the mean of 25, we can find its variance to also be 25.
Because
Therefore, since we are given the mean of 25, we can find its variance to also be 25.
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Let us suppose you are a waiter. You work your first four shifts and receive the following in tips: (1) 20, (2) 30, (3) 15, (4) 5. What is the mean amount of tips you will receive in a given day?
Let us suppose you are a waiter. You work your first four shifts and receive the following in tips: (1) 20, (2) 30, (3) 15, (4) 5. What is the mean amount of tips you will receive in a given day?
The answer is 17.5. Simply take the values for each day, add them, and divide by the total number of days to obtain the mean: 
The answer is 17.5. Simply take the values for each day, add them, and divide by the total number of days to obtain the mean:
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Robert's work schedule for next week will be released today. Robert will work either 45, 40, 25, or 12 hours. The probabilities for each possibility are listed below:
45 hours: 0.3
40 hours: 0.2
25 hours: 0.4
12 hours: 0.1
What is the mean outcome for the number of hours that Robert will work?
Robert's work schedule for next week will be released today. Robert will work either 45, 40, 25, or 12 hours. The probabilities for each possibility are listed below:
45 hours: 0.3
40 hours: 0.2
25 hours: 0.4
12 hours: 0.1
What is the mean outcome for the number of hours that Robert will work?
We are required to find the mean outcome where the probability of each possible result varies--the random/weighted mean. First, multiply each possible outcome by the probability of that outcome occurring. Second, add these results together.
We are required to find the mean outcome where the probability of each possible result varies--the random/weighted mean. First, multiply each possible outcome by the probability of that outcome occurring. Second, add these results together.
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There are 
collectable coins in a bag. 
are 
ounces, 
are 
ounces, 
are 
ounces, and 
are 
ounces. If one coin is randomly selected, what is the mean possible weight in ounces?
There are
We are required to find the mean outcome where the probability of each possible result varies--the random/weighted mean.
First, multiply each possible outcome by the probability of that outcome occurring.
Second, add these results together.
We are required to find the mean outcome where the probability of each possible result varies--the random/weighted mean.
First, multiply each possible outcome by the probability of that outcome occurring.
Second, add these results together.
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A basketball player makes 
of his three-point shots. If he takes 
three-point shots each game, how many points per game does he score from three-point range?
A basketball player makes
First convert 
.
The player's three-point shooting follows a binomial distribution with 
and 
.
On average, he thus makes 
three-point shots per game.
This means he averages 12 points per game from three-point range if he tries to make 10 three-pointers per game.
First convert
The player's three-point shooting follows a binomial distribution with
On average, he thus makes
This means he averages 12 points per game from three-point range if he tries to make 10 three-pointers per game.
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Tim samples the average plant height of potato plants for his science class and finds the following distribution (in inches):
Which of the following is/are true about the data?
i: the mode is 
ii: the mean is 
iii: the median is 
iv: the range is 
Tim samples the average plant height of potato plants for his science class and finds the following distribution (in inches):
Which of the following is/are true about the data?
i: the mode is
ii: the mean is
iii: the median is
iv: the range is
Analyzing the data, there are more 6s than anything else (mode), the median is between 
and 
, the mean is 
, and the range is 
Analyzing the data, there are more 6s than anything else (mode), the median is between
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During a week's worth of soccer practice, a player practices 
total free kicks and has a 
chance of scoring. What is the probability that he or she scored at least 
times? Assume each shot is independent.
During a week's worth of soccer practice, a player practices
Two steps are crucial here.
First, we need to recognize this is a binomial distribution with 
and 
.
Second, we need to realize we can use a normal approximation of the binomial since 
and 
, which are both larger than 5.
With that said, we can calculate a 
-score and its 
-value, keeping in mind that our mean will be 
and our standard deviation will be 
, which is about 
.
Two steps are crucial here.
First, we need to recognize this is a binomial distribution with
Second, we need to realize we can use a normal approximation of the binomial since
With that said, we can calculate a
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Which of the following would be considered a binomial experiment?
Which of the following would be considered a binomial experiment?
There are four conditions that need to be satisfied for a binomial experiment:
-
Each trial must have two outcomes.
-
Each trial must be independent.
-
All trials must be identical.
-
The probabilities of the outcomes remain constant must not change with each trial.
The only choice that satisfies all four of these conditions (and is therefore a binomial experiment) is the rock-paper-scissors scenario.
There are four conditions that need to be satisfied for a binomial experiment:
-
Each trial must have two outcomes.
-
Each trial must be independent.
-
All trials must be identical.
-
The probabilities of the outcomes remain constant must not change with each trial.
The only choice that satisfies all four of these conditions (and is therefore a binomial experiment) is the rock-paper-scissors scenario.
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Suppose you are throwing three darts and you have a one third chance of hitting the bull's eye. Each throw is independent of one another. What is the chance of hitting the bull's eye at least once?
Suppose you are throwing three darts and you have a one third chance of hitting the bull's eye. Each throw is independent of one another. What is the chance of hitting the bull's eye at least once?
To calculate Prob(at least one bull's eye), we can instead compute one minus the complementary probability, P(no bull's eye).
So we have P(at least one bull's eye)=1-P(no bull's eye).
The chance of getting no bull's eyes is 
.
This means the probability of getting at least one bull's eye is
To calculate Prob(at least one bull's eye), we can instead compute one minus the complementary probability, P(no bull's eye).
So we have P(at least one bull's eye)=1-P(no bull's eye).
The chance of getting no bull's eyes is
This means the probability of getting at least one bull's eye is
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A particle travels left with probability one sixth and right with probability five sixths. Each movement is independent of the others. What is the chance that after three movements, the particle ends up one unit to the right?
A particle travels left with probability one sixth and right with probability five sixths. Each movement is independent of the others. What is the chance that after three movements, the particle ends up one unit to the right?
The movements that this particle can make include: RRL, RLR, LRR.
The chance of getting RRL is 
. This is also the chance of getting any of those movements.
To get the total probability, we can add up the individual probabilities since the events are all mutually exclusive.
Thus, we get the following as the solution.
The movements that this particle can make include: RRL, RLR, LRR.
The chance of getting RRL is
To get the total probability, we can add up the individual probabilities since the events are all mutually exclusive.
Thus, we get the following as the solution.
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If you flip a biased coin, which has a 
chance of being heads and 
of being tails, until you get a head, what is the chance that it takes five flips until you get a head?
If you flip a biased coin, which has a
To calculate this probability, we need to calculate the chance of getting 4 tails and then a head.
Each tail has a prob. of 
and a head is 
, so we multiply 
to the power of 4 (because we need 4 tails) by 
(for the single head).
So the probability is
.
To calculate this probability, we need to calculate the chance of getting 4 tails and then a head.
Each tail has a prob. of
So the probability is
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Which of the following is a discrete random variable?
Which of the following is a discrete random variable?
A discrete variable is a variable which can only take a countable number of values. For example, the number of times that a coin can come up heads in ten flips can only be 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, or 10. Thus, there are a countable number of possible outcomes (in this case 11). This is true for coin flips, but not for caterpillar length, water flow, or rates of return for stocks.
A discrete variable is a variable which can only take a countable number of values. For example, the number of times that a coin can come up heads in ten flips can only be 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, or 10. Thus, there are a countable number of possible outcomes (in this case 11). This is true for coin flips, but not for caterpillar length, water flow, or rates of return for stocks.
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Robert's work schedule for next week will be released today. Robert will work either 45, 40, 25, or 12 hours. The probabilities for each possibility are listed below:
45 hours: 0.3
40 hours: 0.2
25 hours: 0.4
12 hours: 0.1
What is the standard deviation of the possible outcomes?
Robert's work schedule for next week will be released today. Robert will work either 45, 40, 25, or 12 hours. The probabilities for each possibility are listed below:
45 hours: 0.3
40 hours: 0.2
25 hours: 0.4
12 hours: 0.1
What is the standard deviation of the possible outcomes?
There are four steps to finding the standard deviation of random variables. First, calculate the mean of the random variables. Second, for each value in the group (45, 40, 25, and 12), subtract the mean from each and multiply the result by the probability of that outcome occurring. Third, add the four results together. Fourth, find the square root of the result.
There are four steps to finding the standard deviation of random variables. First, calculate the mean of the random variables. Second, for each value in the group (45, 40, 25, and 12), subtract the mean from each and multiply the result by the probability of that outcome occurring. Third, add the four results together. Fourth, find the square root of the result.
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We have two independent, normally distributed random variables 
and 
such that 
has mean 
and variance 
and 
has mean 
and variance 
. What is the probability distribution of the difference of the random variables, 
?
We have two independent, normally distributed random variables
The mean for any set of random variables is additive in the sense that
The difference is also additive, so we have
This means the mean of 
is 
.
The variance is additive when the random variables are independent, which they are in this case. But it's additive in the sense that for any real numbers 
(even when negative), we have
.
So for this difference, we have
.
So the mean and variance are 
and 
, respectively. In addition to that, 
is normally distributed because the sum or difference of any set of independent normal random variables is also normally distributed.
The mean for any set of random variables is additive in the sense that
The difference is also additive, so we have
This means the mean of
The variance is additive when the random variables are independent, which they are in this case. But it's additive in the sense that for any real numbers
So for this difference, we have
So the mean and variance are
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If 
and 
are two independent random variables with 
and 
, what is the standard deviation of the sum, 
If
If the random variables are independent, the variances are additive in the sense that
.
So then the variance of the sum is
.
The standard deviation is the square root of the variance, so we have
.
If the random variables are independent, the variances are additive in the sense that
So then the variance of the sum is
The standard deviation is the square root of the variance, so we have
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Consider the discrete random variable 
that takes the following values with the corresponding probabilities:
with 
with 
with 
Compute the probability 
.
Consider the discrete random variable
Compute the probability
This probability is simple to compute:
We want to add the probability that X is greater or equal to two. This means the probability that X=2 or X=3.
Adding the necessary probabilities we arrive at the solution.
This probability is simple to compute:
We want to add the probability that X is greater or equal to two. This means the probability that X=2 or X=3.
Adding the necessary probabilities we arrive at the solution.
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Consider the discrete random variable that takes the following values with the corresponding probabilities:
with 
with 
with 
with 
Compute the expected value of the distribution.
Consider the discrete random variable
Compute the expected value of the distribution.
The expected value is computed as
for any values of x that the random variable takes.
So we have
The expected value is computed as
for any values of x that the random variable takes.
So we have
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The average number of calories in a Lick Yo' Lips lollipop is 
, with a standard deviation of 
. The calories per lollipop are normally distributed, so what percent of lollipops have more than 
calories?
The average number of calories in a Lick Yo' Lips lollipop is
The random variable 
number of calories per lollipop, so the answer is
or
The random variable
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