Exercises

1.1. Express the following expression in summation notation and verify your answer:

x₁y₄ + x₆y₆ +x₁₁y₈ + x₁₆y₁₀

1.2. Write out and simplify the following expressions:

(a) _i=1Σ³ (3a - 4_i)

(b) [5_i=1Σ⁴ (19 + 11x_i) - 30_i=1Σ⁴y_i] / [_i=1Σ⁴ (33x_i + 57) - _i=1Σ⁴18y_i]

(Hint: Combine all terms in the numerator under one summation sign, and do the same for the denominator. Compare the two obtained expressions and find what should be factored out there for cancellation.)

(c) _i=1Σ³ _j=1Σ⁴(3x_i - 4y_j)

1.3. Let z be a standard normal variable. Find:

P(z>-.93)

1.4. Suppose a mutual fund has an annual rate of return that is approximately normally distributed with mean 6% and variance 49 (the variance is a square measure, not a percent).

Find the probability that the annual return will be between 4% and 10%.

1.5. The below table gives daily deliveries of X parcels to a business office, from a 0 minimum to a 15 maximum, and the number of days each delivery happened during a 100-workday period, i.e., 0 parcels delivered 2 days, 2 parcels 5 days, etc.

(a) Give the probability density function of X in the table:

(b) Compute the expected value of X. Explain its meaning.

(c) Compute the variance and standard deviation of X.

(d) Using answers in (b) and (c), give the expected value and variance of Y=4+5X.

1.6. The following table gives 60 observations on three mutual funds, Y, with the average rates of return X of 10% and 25%.

(a) Give the marginal probability functions of X and Y in the following table:

(b) Compute Cov(X,Y).

(c) Find the conditional probability density function of Y given that X = 10.

(d) Check if X and Y are statistically independent. Then explain the meaning in economic terms.