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Mean and variance of random variables. Continuous Vs. Discrete R.V. Rules for random variables. Diversification Example." &What we did so far& BAt the beginning of the class we learned how to describe data. We learned how to describe the data that we observe using histograms, mean, s.d. etc. Last week we talked about random phenomena and probabilities of various outcomes. This week we will talk about random variables, how to describe what we should observe.CZ0 &?&   81. What s a random variable?DEFINITION: Random variable is a numerical outcome of a random phenomenon. Examples of random variables: Number of heads on two coin tosses Number of dots from a roll of two dice Temperature tomorrow Amount of rainfallTjs s *What will we learn about Random Variables?We will learn how to describe random variables using familiar concepts  mean, standard deviation, median etc. But we will be doing this using assumptions about the distribution of a variable instead of actual data. $ ?2. Random Variables - Example5There are two ways to compute average number of heads we should get when we toss two coins: Repeat the experiment a large number of times and then collect and summarize the data Make some (reasonable) assumptions about probability of each outcome, and derive average number of heads using probability theory 2\" \   <Comparing the sample mean and the mean of a random variable CFor a sample of data x1, x2, x3, x4,..., xn-1, xn, Sample mean is C/ ;Comparing the sample mean and the mean of a random variablerFor a probability distribution The mean of the random variable X is Where pj is the probability of each xjz!)'R,R5 Example 2 Let s say I offer you to play the following game: You give me a dollar. I toss two coins and If it s two tails you get nothing If it s one heads, one tails, I will give you your $1 back If it s two heads I give you $2 back Would you want to play such game?6]"]"6 Example 2 LLet s try to answer the question using what we learned about random variables. You will get $0 in 25% of the tosses, $1 in 50% of the tosses and $2 in 25% of the tosses The mean amount that you win is 0.25*$0 + 0.5*$1 + 0.25*$2 = $1 Because you pay $1 to play, your net winnings=$0'Z'7 Example 3Let s say I offer you to play the following game: You give me $5. I toss two coins and If it s two tails you get $1 If it s one heads, one tails I will give you your $5 back If it s two heads I give you $9 Would you want to play such game?6Ww"Ww" DComparing the sample variance and the variance of a random variable FFor a sample of data x1, x2, x3, x4,..., xn-1, xn, Sample variance is G/CComparing the sample variance and the variance of a random variableXFor a probability distribution Variance is Where pj is the probability of each xjRYZ8,8Discrete vs. Continuous R.V.The variables don t always take on discrete values like 0,1,2  such as the number of heads. Continuous random variables can take on any value in a certain range. Examples of continuous r.v.: temperature, speed, rate of return on investment.H#+ kMore Formal DistinctionProbability Distribution of a discrete random variable X can be described by probability of each possible realization of X Probability Distribution of a continuous random variable Y is described by a density curve. Probability of any event is the area under the curve over a range.H7A9dIntuition Check:What is the probability that your body temperature is exactly 98.6F? Probability that a continuous random variable will be exactly equal to some number a is zero!Z66 &Mean and Variance of a Continuous R.V.@Above formulas are for discrete randome varaibles. Mean and variance of continuous variables are more difficult to compute. So, we won t do it in this class. ,  p"44. Rules for means and variances of random variablesTake any random variable X with X and If you create a new variable Y Then, the mean of this variable is And the variance is x ("  : ( Example 4 On average, you buy 10 gallons of gasoline per week. Assuming that your car drives at constant rate of 20 mpg, How many miles per week do you drive on average? If Standard deviation of your gas consumption is 2 gallons, What is the standard deviation of the number of miles you drive? Z/Example 4 (continued)Gas costs $1.85. How much do you spend on gas per week? What is the standard deviation? If Weekly cost of car insurance is $10, how much does your car cost you per week on average (gas+insurance)? What is the standard deviation of the car cost? Z 42Rules for means and variances of random variables Take any two independent random variables X and Y with X , , Y and If you add them together, you create a new variable W. The mean of this random variable is * (  (  (;$O*Example 4 (continued)=There are two cars and two drivers in a household. On average, Driver 1 drives consumes 10 gallons of gas per week and the Driver 2 consumes 2.5 gallons per week. What is the combined average gas consumption of both drivers? How many gallons of gas will they consume? (assuming that both drive at constant 20 mpg)>>:%Mean number of heads on a single coin(Let s define random variable X as the number of heads on one coin toss. So, we have x1=0 with P(X= x1)=0.5 x2=1 with P(X= x2)=0.5 Mean of X is 0.5V.  ;DExample 5  Mean of the sum of R.VvLet s say we have two coins. Let s call the random variables corresponding to the number of heads on each coin as X1 and X2 Define W=X1+X2 What s W ? Using addition rule for the means,r  %<DExample 6  Mean of the sum of R.VLet s say we have ten coins. Let s call the random variables corresponding to the number of heads on each coin as X1, X2, X3, X4, X5, X6, X7, X8, X9, X10 Define Y=X1+X2 + X2+& + X10 What s Y ? Using addition rule for the means,r %=DExample 7  Mean of the sum of R.V~Let s say we have 217 coins. Let s call the random variables corresponding to the number of heads on each coin as X1, X2, & , X217 Define U=Xi What s U ? Using addition rule for the means,r   %?!Intuition CheckConsider the following gamble: I toss 217 coins and you get a penny for each coin that turns up heads. Would you be willing to bet $1.50? How much money would you be willing to place on such gamble? @"8Random variables  Variances<Take any two independent random variables X and Y with X , , Y and If you add them together, you create a new variable W, then it s variance is   (  (  (<A# CAUTION!!When you are adding random variables, remember to ADD VARIANCES, not standard deviations. Variance is equal to sum of variances Standard deviation is NOT equal to the sum of standard deviations. U,Example 4(continued)There are two cars and two drivers in a household. On average, One driver drives 200 miles per week (s.d. is 20 miles) and the other driver drives 50 miles per week (s.d. is 9 miles). What is the standard deviation of their combined mileage? ,f>KB$DExample 8  Var. of the sum of R.VBack to X  number of heads on 1 coin toss. We found that the mean was 0.5*0 + 0.5*1 = 0.5 What is the variance of X? 0.5*(0-0.5)2 + 0.5*(1-0.5)2 = 0.5*0.25 + 0.5*0.25 =0.25dr ,C%What are the variances ofLW=X1+X2 Y=X1+X2 + X2+& + X10 U=Xi ' D&<Example 9  Portfolio AnalysisYou own two stock. You have $1000 in stock A and $1000 in stock B. Both have average return of 10% per year and standard deviation of 7% What is the expected value (the mean) of your portfolio 1 year from now? Standard deviation?E'(5. Application of R.V. - DiversificationFor stock A, the mean is $1,100 and standard deviation $70 For stock B, the mean is $1,100 and standard deviation $70 Thus, expected value of holding one year from now is $2,200 and the standard deviation is $98.99 True only if stocks are independent.X-\Diversification  the golden rule of finance. //(PTHE MORE STOCKS YOU OWN, THE LESS RISKY IS YOUR INVESTMENT (true only if stocks are independent) Show that the more stocks you own the less risk you take on. 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X& $ @` $` <lT ?P   Xx2$ @` %` <H^ ?P  Xx1$ @` &` <f ?` R Value of X   @``B '` 0o ?`pZB (` s *1 ?`p`B )` 0o ?`0 p0 `B *` 0o ?``0 ZB +` s *1 ?0 ZB ,` s *1 ?P P 0 ZB -` s *1 ?  0 ZB .` s *1 ?0 `B /` 0o ?pp0 H ` 0޽h ? ̙33y___PPT10Y+D=' i = @B +m  0 h$(  hr h S \ P   r h S     H h 0޽h ? ̙33y___PPT10Y+D=' i = @B +m  0 l$(  lr l S , P   r l S  P   H l 0޽h ? ̙33y___PPT10Y+D=' i = @B +m  0 $(  r  S  P   r  S     H  0޽h ? ̙33y___PPT10Y+D=' i = @B +m  0 $(  r  S  P   r  S l    H  0޽h ? ̙33y___PPT10Y+D=' i = @B +9  0 `X(  r  S x P     S x <$ 0   `  c $A ??y` `  c $A ??{  `  c $A !??  l  !`  c $A $??v  ` $H  0޽h ? ̙33y___PPT10Y+D=' i = @B +m  0 $(  r  S ̽ `   r  S     H  0޽h ? ̙33y___PPT10Y+D=' i = @B +}  0 `x$(  xr x S 6=P  = r x S I= = H x 0޽h ? ̙33___PPT10i.#+D=' i = @B +  0 (  r  S pψ P     S pЈ <$ 0   `  c $A ??7  `  c $A ??@ `  c $A ?? d  H  0޽h ? ̙33y___PPT10Y+D=' i = @B +}  0 p8$(  8r 8 S و P   r 8 S ڈ    H 8 0޽h ? ̙33___PPT10i.r+D=' i = @B +y  0 0(  x  c $ P   x  c $    H  0޽h ? ̙33y___PPT10Y+D=' i = @B +  0 `Xp(  x  c $ P     c $h <$ 0     c $A [?? ` 8 [$D 0H  0޽h ? ̙33( ___PPT10.+TeD' n*= @B D[' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*}%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*}%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*+8+0+0 +L  0 4,(  x  c $$    x  c $ p     c $A \?? z  8 \$D 0H  0޽h ? ̙33___PPT10+^D|' = @B D7' = @BA?%,( < +O%,( < +Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*+  0 4,`(  x  c $ P   x  c $ `     c $A ]?? @f8 ]$D 0H  0޽h ? ̙33~___PPT10^+ksDB' i = @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(+y  0 0(  x  c $L P   x  c $$    H  0޽h ? ̙33y___PPT10Y+D=' i = @B +5  0 (  x  c $& P     c $' `<$ 0     c $A ^?? @8 ^$D 0  c $A _??0P 8 _$D 0  c $A `??028 `$D 0H  0޽h ? ̙33UM___PPT10-.+_D' i = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*+y  0 0(  x  c $T2 P   x  c $,3    H  0޽h ? ̙33y___PPT10Y+D=' i = @B +}  0 X$(  Xr X S 9 P   r X S :    H X 0޽h ? ̙33___PPT10i.u4+D=' i = @B +y   0  0(  x  c $G `P   x  c $tH    H  0޽h ? ̙33y___PPT10Y+D=' i = @B +    0 @(  x  c $8M P     c $8N <$ 0     c $A ?? Q P 8 $D 0  c $A ?? %. 8 $D 0  c $A ??` @ 8 $D 0H  0޽h ? ̙33 ___PPT10.+>+D' i = @B Du' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(+y  0 P0(  x  c $8Y P   x  c $Z    H  0޽h ? ̙33y___PPT10Y+D=' i = @B +y  0   0(   x   c $` P   x   c $c  0   H   0޽h ? ̙33y___PPT10Y+D=' i = @B +}  0 0h$(  hr h S `4*P  * r h S P0 * H h 0޽h ? ̙33___PPT10i.y.+D=' i = @B +N 0 0$(  $X $ C     $ S L  @   Remember induction  we want to figure out what is true about the whole by looking at the part The process of statistics: We observe a sample. From the sample we get the mean, variance and other statistics The sample staisitcs are random variables  they vary from sample to sample So what we observe is one realization of a random variable The question is how much can we learn about the actual mean  the mean of the random variable Distribution vs. Probability Distribution observed variable versus random variable -- Nz:Wz:*,  (H $ 0޽h ? ̙33 0 @,&(  ,X , C     , S   @   (H , 0޽h ? ̙3380___PPT10.oGwY 0 P0i(  0X 0 C     0 S   @   k3use the formula for the mean of random variable . 43 H 0 0޽h ? ̙3380___PPT10.o* 0 `4:(  4X 4 C     4 S P  @   <In the industry examples I converted random phenomenon into a random variable. I converted a coin toss into how much money (a numerical outcome) you will win when you toss the coins . During the last couple of lectures what we talked about is the probability of certain outcomes and events. when we talked about rolling dice and tossing coins we calculated the probability of different events . But now we can actually calculate the average amount of money going to win if you pursue a certain of gambling strategy . At first sight it might appear to you that there was no application to probability theory in real life besides the world of gambling , but that is not true . For example you can model the number of customers as a random events and if you know how many customers can go to purchase your products you can calculate the average amount of profits that you going to make in your business. 8    j  l       H 4 0޽h ? ̙3380___PPT10.ppd* 0 zr@ (  @X @ C    r @ S   @   H @ 0޽h ? ̙3380___PPT10.spn 0 D~(  DX D C     D S lˑ  @   lThis is the same statement as before. The only thing that is different here is presentation of the questionsH D 0޽h ? ̙3380___PPT10.s! 0 "H(  HX H C     H S Pґ  @   bWhat we accomplished is converted a seemingly Complicated gamble into a simple random variable . 6c.!  H H 0޽h ? ̙3380___PPT10.s& 0 skL(  LX L C    k L S ّ  @   Return=(Price(tomorrow)-Price(Today))/price(today)=1-ratio Price(tomorrow)=Price(today) + Pricetoday*r (random) known known Random This fits the formula for the mean! MEAN Price(tomorrow)=Price(today) + Pricetoday*MEANr a=price today b=price today S.D. Price tomorrow=b S.D. return Plug it in: Stock a: mean port=1,100 Stock b: mean port=1,100 Stock a: S.D port=70 Stock b: S.D port=70 70:100 Var portfolio= var A+Var B=490+490 S.D Portfolio= 99 99:200 DIVERSIFICATION!!! P)   3   =H L 0޽h ? ̙3380___PPT10.soA, 0 |`(  `X ` C    | ` S 0  @   41H ` 0޽h ? ̙3380___PPT10.uza% 0  d*(  dX d C     d S 0: @   ,CHECK IT YOURSELF!!!!!!!H d 0޽h ? ̙3380___PPT10.v eg5$ 0 E(  X  C    !   S HF!  @  !  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BNd[>'#֧C#TCl;L_P_j,%3T2lf8(~l+%mt<h6JI\V6Gs/}x q #=b nc]# 9X@'Φ BFxL`)8::5 339Asdxaaύ&7b:k*}SӚV6my`H_?6,zܯF“۽0' m3X\pޫ|ŗ7s̝`^&'/qP yd+*GCV k865+ y>A7;ֺӾ!K&U_oagv6Z7+hjozTc!A^,>Mǝsy o~TPm.?Wu?%_aWv{hr_Y?'f;0gRePF&.Q`3p=aߚӂn`w{O :.1aGי16'cjEvHܮf+lIipB>k6N3Av:h-_oOl xY]lY>3vw4-l;ݥ{uUxA /L ӴT5HBbW EQ D ,ha84 `瞹{⭾&8CZu53mVk?U!4*-ͮ- ;#6C %ڒ nx]]>+_kbRX<-Stv!G<$+;yile;p9AX~-1gGVODDf(VŠ(s:UQ):DPOgf'L_E8苡`ơ s ab,">"] Zza!]ҴPʭiDFH4)*H" UI (I-\$UUvohRvQCr&ϕWBEpD2Nӱ? !QHsw\);XZU{9Gk/:JU2):_k}D}]_I4pz#TzÊ1UfZLMgٕQbW(#,%WsO|,<,#'2㥛 ymcTW;4e䋕G#'ݿG|Q vȆS7#'㋼6#s%=Z=#:r>Á4/i Zyr菼n%CդzL6J>UzQh/B~^/#ۦ/QwXԼr^ Ϟ7]aW2>^]^V{,A1xj_7mXϛf"S;3H%bW%nzk5=ŷVtW>%gNj,ѽK|JkV~}OBVu7/}3ǃk [rT0_- 0OA0#@E,`JN)>3@DSVPZ]`$Pzdhz,lptxs|a!':@+?pG 9GJL$4&(.0?35UpA:<;H?--[wOh+'0 `h   Random VariablesAlexander Lebedinskywkuuser17Microsoft PowerPoint@ze@n:z@Pl/.Gg  d  y--$xx--'@Times New Roman-.  2 q1?."System:-@Times New Roman-. 2 0+Random Variables.-@Times New Roman-. 2 <.Chapter 4 in PBS.-՜.+,0    $ hOn-screen ShowWestern Kentucky UniversityU#  &Times New RomanDefault Design(Brownstone Equation Editor 5.0 Equation"Random Variables Chapter 4 in PBSPlanWhat we did so far1. Whats a random variable?+What will we learn about Random Variables?2. Random Variables - Example=Comparing the sample mean and the mean of a random variable <Comparing the sample mean and the mean of a random variable Example 2 Example 2 Example 3EComparing the sample variance and the variance of a random variable DComparing the sample variance and the variance of a random variableDiscrete vs. Continuous R.V.More Formal DistinctionIntuition Check:'Mean and Variance of a Continuous R.V.54. Rules for means and variances of random variables Example 4Example 4 (continued)3Rules for means and variances of random variables Example 4 (continued)&Mean number of heads on a single coin#Example 5 Mean of the sum of R.V#Example 6 Mean of the sum of R.V#Example 7 Mean of the sum of R.VIntuition CheckRandom variables Variances CAUTION!!Example 4(continued)#Example 8 Var. of the sum of R.VWhat are the variances ofExample 9 Portfolio Analysis)5. Application of R.V. - Diversification/Diversification the golden rule of finance.  Fonts UsedDesign TemplateEmbedded OLE Servers Slide Titles#_㉛0wkuuserwkuuser  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~Root EntrydO)Pictures,Current UserSummaryInformation(PowerPoint Document(DocumentSummaryInformation8