# Question: The symmetry or skewness lack of symmetry of a distribution

The symmetry or skewness (lack of symmetry) of a distribution is often measured by means of the quantity α3 = µ3/σ3 Use the formula for µ3 obtained in Exercise 4.25 to determine α3 for each of the following distributions (which have equal means and standard deviations):

(a) f(1) = 0.05, f(2) = 0.15, f(3) = 0.30, f(4) = 0.30, f(5) = 0.15, and f(6) = 0.05;

(b) f(1) = 0.05, f(2) = 0.20, f(3) = 0.15, f(4) = 0.45, f(5) = 0.10, and f(6) = 0.05. Also draw histograms of the two distributions and note that whereas the first is symmetrical, the second has a “ tail” on the left- hand side and is said to be negatively skewed.

(a) f(1) = 0.05, f(2) = 0.15, f(3) = 0.30, f(4) = 0.30, f(5) = 0.15, and f(6) = 0.05;

(b) f(1) = 0.05, f(2) = 0.20, f(3) = 0.15, f(4) = 0.45, f(5) = 0.10, and f(6) = 0.05. Also draw histograms of the two distributions and note that whereas the first is symmetrical, the second has a “ tail” on the left- hand side and is said to be negatively skewed.

## Answer to relevant Questions

The extent to which a distribution is peaked or flat, also called the kurtosis of the distribution, is often measured by means of the quantity α4 = µ4/σ4 Use the formula for µ4 obtained in Exercise 4.25 to find α4 for ...Find the moment- generating function of the continuous random variable X whose probability density is given by And use it to find µ'1, µ'2, and σ2. If X and Y have the joint probability distribution f(x, y) = 14 for x = - 3 and y = - 5, x = –1 and y = –1, x = 1 and y = 1, and x = 3 and y = 5, find cov( X, Y). If var(X1) = 5, var(X2) = 4, var(X3) = 7, cov(X1, X2) = 3, cov(X1, X3) = –2, and X2 and X3 are independent, find the covariance of Y1 = X1 – 2X2 + 3X3 and Y2 = –2X1 + 3X2 + 4X3. The manager of a bakery knows that the number of chocolate cakes he can sell on any given day is a random variable having the probability distribution f(x) = 16 for x = 0, 1, 2, 3, 4, and 5. He also knows that there is a ...Post your question