Mixing the Constant with the Normal
Please see the attached file.
There is a minor typo in the part(d).
The function should be 1/(b-a), not 1/(a+b).
From Conditional distribution and the Bivariate Normal distribution.
Please show each step of your solution and check your final answer.
Thank you.
https://brainmass.com/math/probability/mixing-constant-normal-162855
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Solution. By (b), we know that for all
,
where c and d are constants.
By (c), we know that
which is a constant independent of t.
Then by a), b) and c), we can write the conditional probability density function as follows.
................(*)
Note 1): (*) shows that the conditional probability distribution is a normal distribution with mean ct+d and standard deviation as stated in (b) and (a).
Note 2): exp(x) denotes the exponential function
By d), we know that the marginal distribution in X is
So, by a formula for the joint p.d.f.: , we can get the joint p.d.f. as follows. For every real number y,
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