Adding constants to random variables, multiplying random ...

Adding constants to random variables, multiplying random ...

Arithmetic of random variables: adding constants to random variables, multiplying random variables by constants, and adding two random variables together AP Statistics B pp. 373-74 1 Pp. 373-74 are just plain hard

I dont like the way they are written They give you the conclusion, but dont give you a sense of WHY the rule is what it is This lecture gives you the derivation of the rules You do not have to memorize the derivations, but if you understand them, you will understand why the rules are what they are

2 Outline for lecture 3 basic ideas: Adding a constant to a random variable (X+c) Multiplying a random variable by a constant (aX) Adding two random variables together (X+Y) Being able to add two random variables is extremely important for the rest of the course,

so you need to know the rules Once you can apply the rules for X+Y and X+Y, we will reintroduce the normal model and add normal random variables together (go ztables!) 3 Remember! It may be useful to take notes, but this PowerPoint with the narration will be posted on the Garfield web site.

So will a version that does not have narration if you want a smaller file. Different learning: classes like this that make the lectures available on line require different skills than classes where your notes are all you have. 4 Beginning concepts Lets look at the algebra behind

adding, subtracting, and multiplying/ dividing random variables. Here, we will only examine addition and multiplication Subtraction is simply adding the negative of the addend Division is simply multiplying by the reciprocal

of the divisor 5 Adding a constant to a random variable The first thing well try is adding a constant c to a random variable. We will first calculate the mean, and then look at the variance Remember that given the variance,

we can always take its square root and obtain the standard deviation. 6 E(X+c)=E(X)+c, where c=some real number For the next slides, were going to be expanding the series being summed,

and then regrouping the variables and simplifying. 7 Expanding the series Lets expand without the sigma (adding) operator to keep the algebra neater. 8

Rewriting the equation We can rewrite this as a series of individual fractions, since Thus, 9 Regrouping the equation Now, collect like terms:

Note that c/n in parenthesis appears n times Now, rewrite this as a sum: 10 Var(X+c) Var(X+c)=VarX. We start with the basic definition for

variation (VarX): If we have add a constant c on to random variable X, we have Xi+c replacing Xi Remember, the new mean is X+c. 11 Substitute and rewrite the equation So we substitute Xi-c for Xi, and X+c

for X, to get: Lets again deal only with the numerator and expand the square: 12 Quite a mess, right? Look at this:

You wanna simplify THAT????? So lets simplify it by NOT expanding the square. Instead, what is (Xi+c)-(X+c) equal to BEFORE we square it? 13 Simple, simple, simple Distribute the subtraction operator over X+c, and we should get:

Xi+c-X-c=Xi-X If we substitute Xi-X into the numerator, we get our original definition of variation, i.e., 14 What about the standard deviation? The fact that the VARIANCE does not change means the STANDARD DEVIATION does not

change, either. How come? Remember that Since VAR does not change, the standard deviation also does not change when a constant is added to the random variable 15 What have we proven so far? We have looked at the effect of

adding a constant to a random variable X, i.e., using X+c We have 3 conclusions for X+c: X+c=X+c X+c=X Var(X+c)=Var(X) 16 Multiplying a random variable by a constant

Now lets see what happens when we MULTIPLY the random variable X by some constant a Lets look at the mean first: aX. We will substitute aX for X in the definition of the mean: 17 Expand and analyze

Again, lets expand the Xi terms without the sigma: 18 Variance when the random variable is multiplied Seeing what happens with the variance upon multiplication is similar to adding a constant:

19 Again, what about the standard deviation? This derivation also explains why, when we multiply a random variable by a, the standard deviation is a multiple a of the standard deviation of the random variable.

Recall the definition of the standard deviation: 20 Standard deviation: substitute and solve Substitute aX for X, and we get 21

Recap of conclusion for aX (multiplying the random variable by a constant Once again, three conclusions: 22 Final approach: adding two random variables together

Lets substitute in X+Y into our formulae to find out how they change (Remember that X-Y can be recast as an addition problem X+(-Y), so we do not need a separate derivation for X-Y) 23 Calculating the mean when adding two random variables

We again start with the standard definition of the mean, except that we substitute X+Y for X: Once again, calculating the mean is easy peasy. 24 Calculating the variance The variance, of course, will be

harder and messier. In fact, the derivation is so bad that youll have to accept this one on faith: Var(XY)=Var(X)+Var(Y) 25 What about standard deviations? First, lets derive them from the Var formula

Since Therefore: , 26 Recap of adding two random variables together

27 Here endeth the lesson. You are not responsible for these derivations, but I hope it helps to explain why the forms on pp.373-74 are what they are.


Recently Viewed Presentations

  • Urgent Issues in Cyber Security: Preliminary Cyber Security

    Urgent Issues in Cyber Security: Preliminary Cyber Security

    Ron Williamson, Ph.D. Senior Engineering Fellow. Raytheon. ... Relationship with Resiliency and Resilient Systems Engineering. Case Studies on Cyber Attacks. A - Stuxnet ... as we move forward in modernizingour Energy Infrastructure make Cyber Defensea high priority in the engineering...
  • The Insurance Device

    The Insurance Device

    The Insurance Device Chapter 3 Private and Social Private (Commercial) Auto, Life etc. Social (Publicly provided) Social security, Medicare etc Quasi-social FDIC, SIPC, PBGC etc.
  • Post-Classical Asia

    Post-Classical Asia

    Post-Classical Asia. Rise of the Ottoman Empire. The Ottomans, a nomadic group of Turkish people originally from Central Asia . ... Several warring Kingdoms arose and science, art, and culture declined. This period is known as the Six Dynasties.
  • Welcome! []

    Welcome! []

    Orchestra website and handbook (policies) Calendar for the year. Booster Board! How can parents help their children? Questions? Who am I? Mr. Will Whitehead, Director of Orchestras. 2nd year at Braden River High School! Violist. Active with 2 local professional...
  • Diapositiva 1 - CIO

    Diapositiva 1 - CIO

    Puuc architecture has several predominant features, most notably constructions with a plain lower section and a richly decorated upper section. Carvings most commonly found include serpents, lattice work and masks of the god Chac. Chac was the god of rain,...
  • Confirmation Review The Bible Gods Relationship to Man

    Confirmation Review The Bible Gods Relationship to Man

    The Bible. The Old Testament (The Ancient People of Israel) (The Apocrypha) The New Testament (The Life of Jesus and Early Church)
  • School Prayer - University of Utah

    School Prayer - University of Utah

    McCollum v. Bd of Education(1948). Champaign Board of Education offered voluntary religious education classes for public school students from grades four to nine. Weekly 30- and 45-minute classes were led by clergy or lay teachers in public school classrooms during...
  • Industry, Urbanization, Immigration and the Gilded Age

    Industry, Urbanization, Immigration and the Gilded Age

    Horatio Alger "Myth of the Self Made Man" ... Alger's book, Sink or Swim helped . Also Supporter of capitalism. Acres of Diamonds. Conwell. Social Darwinism. Based on the scientific studies of Charles Darwin-Natural Selection. Ideas are applied to society...