Answer:
If we find the expected value of Y we got:
And for the variance we assume that we have independent random variables for X1 and X2 so then we have:
And the deviation
So then our random variable Y have the following distribution:
And we want to find this probability:
And we can use the z score formula given by:
And if we use the z score formula we got:
Step-by-step explanation:
Previous concepts
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".
Solution to the problem
Let X the random variable that represent the lenghts of a population, and for this case we know the distribution for X is given by:
Where and
We slect two observations
And let denote the combined lenght as:
If we find the expected value of Y we got:
And for the variance we assume that we have independent random variables for X1 and X2 so then we have:
And the deviation
So then our random variable Y have the following distribution:
And we want to find this probability:
And we can use the z score formula given by:
And if we use the z score formula we got: