Discrete Random Variables Flashcards
What is a discret random variable and what is a probability distribution
What must the probabilities on the distribution add to
This is a variable for which a KNOWN list of numerical values can be taken, so X can take 1,2,3 etc
If they all have an associated probability of X taking that value, it can be modelled in a probability distribution
The probabilities must add to 1
What is E (X) and Var(X) and how to calculate
E(x) can be though as the LONG TERM average score of results, so the mean
= sum of XP
Var (x) is the variance in data
= E (x2) - ( Ex)2
Where e (x2) is the sum of X2P
How to find standard deviation?
Sqaure root var x
How to use E(x) and Var(x) to compare two data sets?
Comment on the means, showing overall increased etc
Common on the variance, data more spread less spread, so consistent or not etc
Why is the variance formula = to E((X-U))2
How to make this formula alternate form?
(What’s the expected value of a CONSTANG)
Variance is difference of mean and x value, sqaure it, sum it and divide by total (find mean)
Now E(x) is the mean basically, thus saying E( x -u) 2 gives the VARIANCE
2) expand the inside bracket
- expected value of a CONSTANT like mew = constant
- expected value of x remember is the mean = mew
Rearrange
How to find e( aX + b) and why
What about var (ax b)
What about standard deviation?
1) = a E(x) +b
- because mean is multiplied and added
2) a2 E(x)
- because standard deviation, the spread, becomes more spread by factor of a, so variance becomes spread by fsctor a2
What’s going on when we do E(X+Y) what’s happening
We are adding two separate probability distributions up and finding out their probability distribution overall’s expected value
Rules for E(X+-Y)
Why
E(x+-y)= E(x) +- E(y)
Thinking about out, the means are gonna add or subtract accordingly
Can prove by putting it as E(x) + E(-y) and using e(ax) = -E(x)
Rules for VAR (X+-Y)
Why
VAR (X+-Y) = VAr (x) +var (y)
Why? Well no matter if they are adding or subtracted the spread of data is getting more
Again prove as var (ax) = a2 var(x), and here it will become negative 1 sqaured so they always add
REMEMBER WHATS THE RULE FOR DOING ANY VAR(X+-Y)
THEY MUST BE INDEPENDENT
So if 10 independent same trials of one thing that had var 2 happened, what is new var
So if he repeated 10 times, you know they independent
New var is 20!
Remember steps to do any question
1) define each variable
2) write down final function
3) now find each thing individually , don’t MESS UP
Finally do combination equation
Importsnt thing when combining variables what never to do!
Say it’s var (x +x ) NEVER COMBINE TI MAKE VAR (2X), as this would become 4 var x when it should be 2 var x as they are independent
