Download MAT1193 – Notes on parameters and variables Parameters

MAT1193 – Notes on parameters and variables Parameters and Variables Parameters and variables can best be understood by thinking about designing and running an experiment. Suppose we want to find out the time course of action of a drug controlling blood sugar levels. An injection is given, and blood sugar levels are recorded each hour after the injection. In this case, blood sugar level is a variable – call it b -­‐ that depends on time, which is also a variable (call it t). Usually we’d like to find out the function the tells us the blood sugar level as a function of time: b(t). Now suppose we want to determine how the time course of the blood sugar level for different concentrations of the drug in the initial injection. So in applications the distinction between parameters and variables usually depends on the timescale over which they change: variables are usually the things that change within one experiment or experimental trial, while parameters are the values that change across experiments or experimental trials. Here’s a simple example: Suppose we consider the growth of a bacterial colony. Suppose that due to reproduction, each hundred bacteria will produce 10 new bacteria after one day. If we let bt be the number of bacteria on day t, we can write this relationship as bt+1 = bt+ (10/100)*bt or bt+1 = 1.1*bt 1.1 is the per capita production rate – the rate of production of new bacteria per bacterium. Now suppose we grow this bacteria in an environment that has a richer source of food. Then the population will grow more rapidly. So the general equation for our simple model of bacterial growth can be written bt+1 = r*bt where r is the per capita production rate. For a given type of bacteria and a given environment, r will be constant, while the population of the bacteria bt will change from day to day. Note the day t will also change. We call b Here’s another example: Suppose a fish farmer raises fishes, and we let fm be the number of fish at month m (f3 is the number of fish after the third month). Also, the per capita production in the ideal case is 1.5 but decreases as there are more fish due to overcrowding: r = 1.5-­‐fm /1000. Finally, each month the farmer harvests a fixed fraction H of all the fish that are at least one month old. So then the growth in fish from month to month is given by fm+1 = r*fm – H*fm = = (1.5-­‐fm /1000)*fm – H*fm In this formula fm is a variable that changes from month to month and H is a parameter that is fixed. In the long term the fish population will settle down to an equilibrium where the number of new fish being born equals the number of fish harvested. Call that ending number fish F. Then the farmer will get a monthly fish harvest equal to H*F. Now we consider changing the parameter H. If H is too big then he’ll harvest too much per month and the population will not get very big. F will be small so H*F may not be big even if H is near 1. On the other hand if he doesn’t harvest a big fraction of fish, H will be small and so even though f will grow to be a big number, his harvest H*F will be small. What is the harvest fraction H that maximizes H*F? In this problem we need to step back and look at what happens for each strategy for harvesting fish. In this case, the harvest fraction H is our variable and the equilibrium number of fish will depend on that harvest fraction. That is F is a function of H, and we can write F(H). The problem can be restated as finding the harvest fraction H that maximizes the function G(H)=H*F(H). So for the particular situation at a given harvest rate H, H is acting as a parameter and fm is the variable that depends on time. But in the bigger picture, H is the variable that determines the farmer’s strategy, and the final fraction of fish F is the outcome variable. Mathematically, variables are the values that change each time you evaluate a particular function. Specifying parameters allows you to described whole groups of functions. For example L(x)=m*x+b or y=m*x+b. x is the input (or independent) variable, and y is the output (or dependent) variable. Notice just writing the function as L(x) tells us that the output of the function L depends on the input variable x. The parameters m and b are values that are take on fixed values in a particular situation, but can change from situation to situation. For example, m=3 and b=5 specifies the specific function F(x) = 3*x+5 . L(x)=m*x+b is a way of talking about all functions whose graph is a line, and Q(x) = 3*x+b is a way of describing all line functions whose slope is equal to 3.