Question

You draw the following ve numbers from a standard normal distribution:
f= (1:7; 0:55; 0:85)
What are the equivalent draws from a normal distribution with mean 0:8 and variance
25?
b. Suppose x1 ~ N(1; 5) and x2~ N(2; 2). The covariance between x1 and x2 is 1:3.
What is the distribution of x1 + x2 and x1 - x2?
c. Suppose x1 ~N(1; 5), x2 ~N(2; 3), and x3 N(2:5; 7), with correlations P1;2 = 0:3,
P1;3 = 0:1 and P2;3 = 0:4. What is the distribution of x1 + 2x2 - 3x3?

Answer 1

Solution :

(a) Suppose that the following positive numbers are drawn from a standard normal distribution :

We have to find the equivalent draws from a normal distribution with mean 0.8 and variance 25.

We know the fact that if   is a normally distributed random variable, i.e., if ,

The information that is given is ,

Thus , the equivalent draw of   from   will be ,

  

Thus , the equivalent draw of    from   will be ,

  

Thus , the equivalent draw of    from   will be ,

  

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(b) We have ,  

We have to find the distribution of and

Since   are Normally distributed random variables , we know that any linear combination of will also be Normally distributed. Thus , clearly   and   will also be Normally distributed. From the information given , we have ,

Hence , the Expectation and Variance of    will be ,

Thus ,

Hence , the Expectation and Variance of   will be ,

Thus ,

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(c) We have ,  

We have to find the distribution of   

Since   are Normally distributed random variables , we know that any linear combination of will also be Normally distributed. Thus , clearly   will also be Normally distributed. From the information given , we have ,

The Pairwise Correlations among the 3 Normally distributed random variables are ,

The Pairwise Covariances among the 3 Normally distributed random variable will be ,

  

  

  

Hence , the Expectation and Variance of   will be ,

  

Thus ,