How to find standard deviation of a random variable - AP Statistics
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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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