5.3

Weibull exerciseLifetimes in years of an electric motor have a Weibull distribution with b = 3 and a = 5. Find a guarantee time, g, such that 95% of the motors last more than g years. That is, find g so that P(X > g) = 0.95

What is called a linear combination of random variables?

aX+b, aX+bY+c, etc. 2X+3 5X+9Y+10 5.5.3 Means and Variances for Linear CombinationsIf a and b are constants, then E(aX+b)=aE(X)+b.__________________________

Corollary 1. E(b)=b.Corollary 2. E(aX)=aE(X).

Suppose that 20 ohm resistors have a mean of 20 ohms and a standard deviation of 0.2 ohms. Four independent resistors are to be chosen a connected in series. R1 = resistance of resistor 1R2 = resistance of resistor 2R3 = resistance of resistor 3R4 = resistance of resistor 4The systems resistance, S, is S = R1 + R2 + R3 + R4 Find the mean and standard deviation for the system resistance, S.

5.5.4 Propagation of Error Often done in chemistry or physics labs.In last section we talked about the case when U=g(x) is linear in x

When U=g(X,Y,Z) is not linearWe can get useful approximations to the mean and variance of U.Using Taylor expansion of a nonlinear g,

Where the partial derivatives are evaluated at (x,y,,z)=(EX, EY,,EZ).

The right hand side of the equation is linear in x,y,z.

For the mean E(U)since

For Var(U)

Example

Example 24 on page 311

The assembly resistance R is related to R1, R2 and R3 by

A large lot of resistors is manufactured and has a mean resistance of 100 with a standard deviation of 2 . If 3 resistors are taken at random , can you approximate mean and variance for the resulting assembly resistance?

5.5.5 The Central Limit Effect

If X1, X2,, Xn are independent random variables with mean m and variance s2, then for large n, the variable is approximately normally distributed.

If a population has the N(m,s) distribution, then the sample mean x of n independent observations has the distribution.For ANY population with mean m and standard deviation s the sample mean of a random sample of size n is approximately

when n is LARGE.

21Central Limit Theorem If the population is normal or sample size is large, mean follows a normal distribution

and

follows a standard normal distribution.

22The closer xs distribution is to a normal distribution, the smaller n can be and have the sample mean nearly normal.

If x is normal, then the sample mean is normal for any n, even n=1. Usually n=30 is big enough so that the sample mean is approximately normal unless the distribution of x is very asymmetrical.

If x is not normal, there are often better procedures than using a normal approximation, but we wont study those options.

Even if the underlying distribution is not normal, probabilities involving means can be approximated with the normal table.HOWEVER, if transformation, e.g. , makes the distribution more normal or if another distribution, e.g. Weibull, fits better, then relying on the Central Limit Theorem, a normal approximation is less precise. We will do a better job if we use the more precise method.

ExampleX=ball bearing diameterX is normal distributed with m=1.0cm and s=0.02cm =mean diameter of n=25Find out what is the probability that will be off by less than 0.01 from the true population mean.

ExerciseThe mean of a random sample of size n=100 is going to be used to estimate the mean daily milk production of a very large herd of dairy cows. Given that the standard deviation of the population to be sampled is s=3.6 quarts, what can we assert about the probabilities that the error of this estimate will be more then 0.72 quart?Let Means of Linear Combinations

ExampleX, Y, Z = values of three diceYou win dollars

Your total expected winnings are

Expected value (constant) = constant

E

Variances of Linear Combinations

when X, Y and Z are independent.

Adding a constant doesnt change variance

Variances are in units2.

If X, Y, Z are independent or uncorrelated.

Putting these facts together

For independent X, Y

Like Pythagorean Theorem, Most useful facts:If X1, X2, X3, , Xn are independent measurements each with and then To see this:

The sample mean is an unbiased estimator of the population mean .

If X1, X2, , Xn are normal, then is exactly normal.Example: Bags of potatoes weigh

If we buy 4 bags, what is ?