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Sometimes we need to combine two random variables For example, suppose one contractor can finish a particular job, on average, in 40 hours (mx = 40) Another contractor can finish a similar job in 35 hours (my = 35) If they work on two separate jobs, how many hours, on average, will they bill for completing both jobs It should be clear that the average of X + Y is just the average of X plus the average for Y That is,

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Calcium is taken up by plant roots either directly from the soil or from groundwater When calcium is extracted from the soil via membrane permeability, both active and passive

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transport of ions takes place Calcium is then transferred and stored in the leaves for a time Calcium returns to the soil when leaves fall It is stored in woody plant parts until it decays, is burned, or is consumed by an animal

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The situation is somewhat less clear when we combine variances In the contractor example above, suppose that

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Nitrogen (N2) makes up 79% of the atmosphere All life on Earth requires nitrogencontaining compounds (eg, proteins) to survive However, they can t easily use nitrogen in its gaseous form To be used by a living organism, nitrogen must be combined with hydrogen and oxygen Nitrogen is pulled from the atmosphere by lightning or nitrogenfixing bacteria During storms, large amounts of nitrogen are oxidized by lightning and mixed with water (rain) This falls and is converted into nitrates Plants take up nitrates and form proteins Plants are consumed by herbivores or carnivores When these consumers die (organic matter), nitrogen compounds are broken down into ammonia Ammonia can be taken up by plants again, dissolved by water, or remain in the soil to be converted to nitrates (nitrification) Nitrates stored in soil can end up in rivers and lakes through runoff It can also be

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Precipitation Nitrogen loss Fertilizers Runoff Organic biomass Lake Ammonium Nitrates Leaching Nitrification Nitrites

SOAPS, COSMETICS, AND PERSONAL-CARE PRODUCTS (continued) Maybelline Inc Soaps, Cosmetics and Personal-Care Products Dec-95 Nov-91 Dec-90 Apr-97 Jan-92 Mar-95 Jun-90 Soaps, Cosmetics and Personal-Care Products Food and Related Products Transportation Equipment Soaps, Cosmetics and Personal-Care Products Wholesale Trade, Nondurable Goods Electronics and Electrical Equipment Mennen Co BSN-Lanson, Pommery Champagne Stant Corp (Bessemer Capital Partners) Barnangen (Nobel Industrier) Celine(Au Bon Agache) Phillips Industries Inc

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changed into free nitrogen and returned to the atmosphere Figure 104 gives you an idea of the nitrogen cycle

Does the variance of the sum equal the sum of the variances Well, yes and no Yes, if the random variables X and Y are independent (that is, one of them has no influence on the other, ie, the correlation between X and Y is zero) No, if the random variables are not independent, but are dependent in some way Furthermore, it doesn t matter if the random variables are added or subtracted, we are still combining the variances That is,

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