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Business Mathematics II
Project 2: Variance
CLASS ACTIVITY
Finite Random Variables If X is a finite random variable, then the variance and standard deviation of X are given by
ᡈ䙦ᡐ䙧 㐄 ∑ (^) ⤑⤢⤢ け䙦ᡶ ㎘ † 〥䙧⡰^ ᙨ ᡘ〥䙦ᡶ䙧 and …〥 㐄 㒓ᡈ䙦ᡐ䙧 , respectively.
- The p.m.f. of a finite random variable, X , is given below.
x 1 2 4 8 ᡘ〥䙦ᡶ䙧 0.05^ 0.15^ 0.30^ 0.
(a) Compute ᡈ䙦ᡐ䙧.
x ↈↀ䙦∆䙧 ∆ ᙨ ↈↀ䙦∆䙧 䙦∆ ㎘ ≆ↀ䙧❹^ ᙨ ↈↀ䙦∆䙧 1 0. 2 0. 4 0. 8 0.
ᡈ䙦ᡐ䙧 㐄 㔳䙦ᡶ ㎘ †〥䙧⡰^ ᙨ ᡘ〥䙦ᡶ䙧
⤑⤢⤢ け
(b) Compute …〥.
If X is a Bernoulli random variable with parameter p, then ᡈ䙦ᡐ䙧 㐄 ᡨ ᙨ 䙦1 㐄 ᡨ䙧 and …〥 㐄
㒓ᡨ ᙨ 䙦1 ㎘ ᡨ䙧.
- Let Y be a Bernoulli random variable with parameter ᡨ 㐄 0.8.
(a) Use the general formula for the variance of a finite random variable to compute ᡈ䙦ᡑ䙧. (b) Use the special formula for the variance of a Bernoulli random variable to compute ᡈ䙦ᡑ䙧. (c) Compute …〦.
If X is a binomial random variable with parameters n and p, then ᡈ䙦ᡐ䙧 㐄 ᡦ ᙨ ᡨ ᙨ 䙦1 ㎘ ᡨ䙧 and
…〥 㐄 㒓ᡦ ᙨ ᡨ ᙨ 䙦1 ㎘ ᡨ䙧.
- A company has a force of 10 salespeople, whose sales performances can be considered to be independent of each other. The manager estimates that the probability of any one sales person reaching his or her monthly quota is 0.8. Let Q be the number of salespeople who reach their quotas in a given month. Then Q is a binomial random variable with ᡦ 㐄 10 and ᡨ 㐄 0.8.
(a) Use the general formula for the variance of a finite random variable to compute ᡈ䙦ᡃ䙧. (b) Use the special formula for the variance of a binomial random variable to compute ᡈ䙦ᡃ䙧. (c) Compute …〘, and interpret the result in terms of the number of salespeople who reach their quotas in a given month.
Continuous Random Variables If X is a continuous random variable, then the variance and standard deviation of X are given by
ᡈ䙦ᡐ䙧 㐄 ᔖ⤑⤢⤢ け 䙦ᡶ ㎘ † 〥䙧⡰^ ᙨ ᡘ〥䙦ᡶ䙧 ᡖᡶ and …〥 㐄 㒓ᡈ䙦ᡐ䙧 , respectively.
- Let X be the random variable that gives the fraction of our company’s revenue that is generated by online sales at a randomly selected time during the coming year. A sales manager suggests that the p.d.f. and c.d.f. for X are given by
⡱ (^) ㎗ 12 ᙨ ᡶ⡰ (^) if 0 㐉 ᡶ 㐉 1 0 elsewhere
and
⡲ (^) ㎗ 4 ᙨ ᡶ⡱ (^) if 0 㐉 ᡶ 㐉 1 0 elsewhere
,^3
respectively.
(a) Set up an integral that gives ᡈ䙦ᡐ䙧, and use Integrating.xlsm to evaluate the integral. (b) Compute …〥, and interpret the result in terms of the fraction of our company’s revenue that is generated by online sales.
Samples If 䙨ᡶ⡩, ᡶ⡰, … , ᡶぁ䙩 is a random sample from a random variable X, then the sample variance and
sample standard deviation are given by ᡱ⡰^ 㐄 ⡩ ぁ⡹⡩ ᙨ ∑ ぁ〶⢀⡩䙦ᡶ 〶 ㎘ ᡶᆑ䙧⡰ and ᡱ 㐄 √ᡱ⡰ , respectively.
- Let X be the number observed on one toss of a fair octahedral (8-sided) die. One thousand samples of size ᡦ 㐄 5 observations of X are given in the sheet Simulation of Dice 2.xlsx.
(a) Use the formulas for the mean, variance, and standard deviation of a sample to compute ᡶᆑ, ᡱ⡰, and ᡱ for the first sample. (b) Use the AVERAGE , VAR , and STDEV functions in Excel to compute ᡶᆑ, ᡱ⡰, and ᡱ for each of the 1,000 samples.
If ᡶᆑ is the sample mean for samples of size n from any random variable X, then ᡈ䙦ᡶᆑ䙧 㐄 ᡈ䙦ᡐ䙧/ᡦ
and …けᆑ 㐄 …〥/√ᡦ , respectively.
- Let X be a random variable with a mean of 60 and a standard deviation of 7.5, and let ᡶᆑ be the sample mean for random samples of size ᡦ 㐄 25. Compute the expected value, variance, and standard deviation of ᡶᆑ.
