(a)A random variable defined as "/36 (Zx - 1); X = 1,2345,6 FonsmnicFiohability distrihation function ofXPcX=The table ahov gives X, the randor VariableManthus...

The following table provides a probability distribution for the random variable $x$ .
$\begin{array}{cc}{x} & {f(x)} \\ {3} & {25} \\ {6} & {50} \\ {9} & {.25}\end{array}$
a. Compute $E(x)$ , the expected value of $x$
b. Compute $\sigma^{2}$ , the variance of $x .$
c. Compute $\sigma,$ the standard deviation of $x .$

In this problem we are finding the expected value, the variants and the standard deviation of X. So let's start with the expected value. The expected value is what you would expect on average to get. If you play this game an infinite number of times. And so in order to do that we are going to multiply each of the outcomes by the respective probabilities and sum them up. And so in this table we have three possible outcomes. So the first one is one And the probability of that happening is four nights plus outcome two. The probability of that is also four nights Plus, the probability of three happening is one night. So we can go ahead and multiply these. So 4/9 plus 8/9. So 12 9th plus three nights is 15 Nights and 15 divided by nine is one In 1/3 or 1.67. So when you're playing this game you either get an outcome of one or two or three. Um but if you play this game, an infinite number of times the average outcome that you get is 1.67. So you are getting twos and threes way more often than your, excuse me, you're getting ones and two is way more often than you're getting threes and that's also evident by the probabilities that were given next is the variance. The variance is calculated by doing the outcome. So in this case one- the expected value, which is 1.67 quantity squared Times the probability of that outcome, which was four nights. And then repeating this process for all three outcomes. Using my calculator for this one, I got point 1995 plus For this one, I got .0484. And let's see the 3rd 1 For the last one, I got .1965 adding all those together, I got a variance of .4444 and then last but not least. The standard deviation Is just the square root of the variance, so the square of .4444. And if you wanted to solve for that, numerically, I got .6666, there we go.

In this problem, we're going to be finding the expected value of the variance and standard deviation using the table provided. So let's go ahead and start with expected value. When we calculate the expected value, we're multiplying the outcome times the probability of that outcome happening and then adding up the some all the way through the chart. This chart only has two possible outcomes and so we're going to have The outcome of zero times 1 5th Plus the outcome of one times 4/5. And so the expected value is going to be forfeits. So that's basically saying you get one sometimes you get zero other times but you get one way more than you get zero. So the expected outcome is 4-5 or .8. The variance is calculated by doing the possible outcome minus the expected value quantity squared times the probability of that outcome. Just one thing. And then we add that to the second possible outcome. So the second outcome is 1- the expected value quantity squared Times the probability of the outcome of one which is forfeits. Yeah. Using my calculator, I get 4/5 quantity squared are negative forfeits, quantity squared which is a positive number times 1/5 And I got 16 out of 125 plus now one minus 4/5 which is a positive 1/5 quantity squared is one 25th times for right plus four Out of 1 25. And so the variance is 20 Out of 1 25. Last but not least. The standard deviation is the square root of the variance. So that's going to be the square root 20 Over 1 25 and that's it.

Given mean and standard deviation values. Photo independent around available sex and why we'll find a radio signal square. What exits for why it's 25. So first we have to find the minister in a division for three. EDS. So e 03 s is equal to train toe your fix that is equal toe drink do then that is 30 our Sigma three x z Quito Square Rudolph Variants of three X that is squared off nine to wait until franks as wagons off air to equal. Do a square, wait until fakes so that this training, too it's quiet or don't millions is do Do not be six. Next, we'll find school lightless six. Yours by plus six. Musical. Do you know why you're six New York wise Given us 20 let's six. It will be 26. Next, we'll find the rape standard Deviation Y plus six that is, equality. Scram to very until quite listless, billions off explicit si X plus or minus C. This will do billions off heads. So by that truly we get millions of five less sexy Burstow Brady himself. My so that would be the standard deviation of why that is right. Next we have to find four X plus y. Since X and Y are independent, you fix place. Why sequel? Toe your eggs. This you who are that is equal toe 10 plus 20. So Sigma X, this right is equal to square toe Brilliant self explosive are given X and Y are independent So we have squared our door building So fakes place buildings of life nuggets Great. Who place 25 That is he Quito by a point 3852 Next we have to find four x minus way you explain us. Why is equal to your fix minus feel? Why that IHS me Quinto then minus 20 that is minus 10. Understand the deviation. Sig mawf X minus y we'll square. You do Maybe until X minus flight on. We know that radiant self X plus or minus y is equal to millions off X plus variant off. Why therefore weekend? I does us They're angel fixed less ratings of flight. So those four yes can defy the square root of training and is equipped five point Create five to approximately next we have to find for X one plus x two and we know the random 1,000,000 random variables are independent. Your ex one place X two is a photo. Your fix one place your ex too. That is acquittal 10 plus 10 there was doing, Do you think? More X one place X to the equator. Great talk, Canadian. So x one bless xto that is equal to square photo. Ready until fix one. Bless. Bearing. So her x two that's squared off. Hopeless. Who that is two point. Hey, Too grateful.

All right. So here we're given a random variable with its probability distribution and asks us some questions about it. So for party, it wants us to find the expected value of this variable. So a at once expected value of X, which could also be written as mu of X and recall the move X equals the sum of each value. The random variable can take times f of X. The probability of it occurs. It's a weighted average. So looking at the chart, we see that you have X equals two times 0.2. Plus, there's a value of four 0.3% of the time. Plus it has a value of seven 0.4% of the time. And finally it is a value of eight, the remaining 0.1 of the time. So if you plug this into a calculator, you will find that this is 5.2. All right, So for part B, it wants us to find the variance of X, which can also be written as the standard deviation squared of X, and that is equal to have some of each of the variables values distance from the mean squared waited by the probability. So this works out to if we look at the chart again to minus 5.2 squared times 0.2 plus four minus 5.2 squared times 0.3 plus seven minus 5.2 squared times point for plus a minus 5.2 squared times 0.1. And that works out to If you have a calculator, it makes us much easier. Works out to four point 56 All right. Another last part part, See, wants the standard deviation of acts, and this is just equal to the square root of the variance. Well, charity calculated. So the standard deviation equals the square root, but 4.56 which equals about two 0.14 And if you have a T I 84 calculator, there's a really easy trick to do. Both of these just from the calculator. So put, I'll just do ah, calculator trip. Okay, so if you have a t I 84 put X into list one and put F of X into l two and make sure they're lined up properly. Then do one of our stats. L one then l, too, as the frequency list and it will tell you the mean and the standard deviation, which is really cool and saves a lot of time.