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P1.T2.205 Sampling distributions (Stock & Watson)

David Harper CFA FRM

David Harper CFA FRM
Staff member
AIM: Describe the key properties of the normal, standard normal, multivariate normal, Chi-squared, Student t, and F distributions.


205.1. Two variables each have a normal distribution: X = N(20,4) and Y = N(40,9). The correlation between X and Y is 0.20. Let the (J) be a bivariate normal distribution such that J = 20*X + 28*Y. What is the Pr(1,355 < J < 1,753)?

a. 90.0%
b. 93.0%
c. 94.0%
d. 96.0%

205.2. Each of the following is true about the student t distribution EXCEPT:

a. The student t distribution has skew equal to zero; variance equal to df/(df - 2) where (df) is degrees of freedom; and kurtosis greater than 3.0 (leptokurtosis with heavier tail and higher peak compared to the normal)
b. To test of significance of a single partial slope coefficient in a (sample) multiple regression with three independent variables (aka, regressors), we use a critical t with degrees of freedom (d.f) equal to the sample size minus four (n - 4)
c. The student's t distribution is the distribution of the ratio of a standard normal random variable divided by the square root of an independently distributed chi-squared random variable with (m) degrees of freedom divided by (m)
d. For asset returns involving large sample sizes (for example, n = 1,000), the student t should be used to simulate heavy tails as asset returns exhibit heavy tails

205.3. Each of the following is true about the chi-square and F distributions EXCEPT:

a. The chi-square distribution is used to test a hypothesis about a sample variance; i.e., given an observed sample variance, is the true population variance different than a specified value?
b. As degrees of freedom increase, the chi-square approaches a lognormal distribution and the F distribution approaches a gamma distribution
c. The F distribution is used to test the joint hypothesis that the partial slope coefficients in a multiple regression are significant; i.e., is the overall multiple regression significant?
d. Given a computed F ratio, where F ratio = (ESS/df)/(SSR/df), and sample size (n), we can compute the coefficient of determination (R^2) in a multiple regression with (k) independent variabels (regressors)