Do bonds reduce the overall risk of an investment portfolio? Let x be a random variable representing annual percent return for Vanguard Total Stock Index (all stocks). Let y be a random variable representing annual return for Vanguard Balanced Index (60% stock and 40% bond). For the past several years, we have the following data. x: 21 0 12 11 31 21 11 −14 −17 −8 y: 8 −2 26 14 15 21 13 −2 −2 −2 (a) Compute Σx, Σx2, Σy, Σy2. Σx Σx2 Σy Σy2 (b) Use the results of part (a) to compute the sample mean, variance, and standard deviation for x and for y. (Round your answers to four decimal places.) x y x s2 s (c) Compute a 75% Chebyshev interval around the mean for x values and also for y values. (Round your answers to two decimal places.) x y Lower Limit Upper Limit
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
Do bonds reduce the overall risk of an investment portfolio? Let x be a random variable representing annual percent return for Vanguard Total Stock Index (all stocks). Let y be a random variable representing annual return for Vanguard Balanced Index (60% stock and 40% bond). For the past several years, we have the following data.
| x: |
21
|
0
|
12
|
11
|
31
|
21
|
11
|
−14
|
−17
|
−8
|
| y: |
8
|
−2
|
26
|
14
|
15
|
21
|
13
|
−2
|
−2
|
−2
|
| Σx | Σx2 | ||
| Σy | Σy2 |
(b) Use the results of part (a) to compute the sample mean, variance, and standard deviation for x and for y. (Round your answers to four decimal places.)
| x | y | |
| x | ||
| s2 | ||
| s |
(c) Compute a 75% Chebyshev interval around the mean for x values and also for y values. (Round your answers to two decimal places.)
| x | y | |
| Lower Limit | ||
| Upper Limit |
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