Chapter 2) Discrete Distrubutions : Uniform + Geometric Flashcards
What’s is a uniform discrete distribution
Thus what is P (X=r)
Where all the probabilities of getting x is the same
Thus P(x=r) = 1/n
What is e(x) and var (x) for uniform
E(x) = n+1/2 (median )
Var (x) = n2-1/12
How to easily remember e(x) for uniform?
Remember dice situation, it was 3.5 which was median
How to derrive e(x) and var(x) for anything
Remember standard results for summations?
E(x) is just sum from 1 to n of r x p
Var(x) = e(x2) - (e(x))2
Meaning fir var x find out the sum of 1 to n if r2 x p
Now apply these formulas , use standard results for summations, and factorise!
However standard results for E(x) and var(x)fir uniform are defined for n starting 1
What if you’re sequence off values DINT START FROM 1?
- what must find first
basically can think of a linear transformation applied to E(x)
SO FIRST FIND THE NTH TERM, AND PUT THIS INTO E(X)
So like 6,12,18 (must be linear)
Becomss E(6x)
So 6ex and 36varx
APPLY prior knowledge!
If you are trying to find uniform probability for a range like 10< x <20 WHAT TO DO EACH STEP
1) discrete form?
2) what is probability of 1 (how to find in discrete distribution what to watch out for)
3) how many numbers in range
Final andwer
1st step always make it discrete form
= 11<= x<= 19
Now need to know what n is, as probiloty is 1/n
- be careful don’t slip up, n might not just be the LAST NUMBER, n is defined as total numbers in the list!
- now need to find the range if numbers in the range
This is always 19-11 +1 as its 11 INCLUDED!
Finally multiply probability by number as they will all add up
Again if the range is 11<= x <= 19 what is the RANGE BETWEEN of numbers (how many numbers)
Think from 1 to 10, there are 10 not 9, so here will be 9 too!
But if the range starts at 1, and ends at n, how many numbers do you have?
Yiu have n numbers if it starts at 1
If it doesn’t, must subtrsvt and ADD 1
What is the gerometric distribtuin all about
Thus what is the p (x=r)
A special case of binomial, where you want each trial except the LAST one to lose,
So los Los lose win
P(x=r) = (1-p)^r-1 x p
In terms if conditions for geometric, what are they, and what’s the difference with binomial?
- indepent
- only two possibilities, success failure
- probabilities are FIXED
HOWEVER N IS NOT FIXED, IT CAN BE ANY NUMBER REMEMBER AS YOU JUST MAKE THE LAST TRIAL A WIN!
Why do we not normally use a poission for modelling geometric cases?
This is because they aren’t RANDOM , happening all at once sometimes
- thus not uniform average mean either
How to find P x >3
This essentially means LOSING THREE TIMES IN A ROW, before even trying to win after
So must lose , q ^3
How to thus find x <=3 geometric
This means 1 - P x>3
So 1 - q ^3
What is E(x) and Var(x) geometric
1/0 and 1-p/02
What does e(x) in geometric neben mean
It means the mean shots before a SUCCESSFUL SHOT TAKEN
What’s ALWAYS the highest probability of hitting a shot in any geometric distribution, Joe many attempts?
Always 1, because decimal keeps getting lower lower
So 1 shot is the highest chance he’ll hit it ,
When there is given conditional probability for gemtreic what happens
It doesn’t matter what it was before, if itsprob =10 given lost 7, it just ,sans prob of getting 3 bevause7 before have no effect
The intercept between greater 7 and 10 is just 10, so even after you should be able to work it out
HOW TO FIND ANY RANDOM SEQUENCE VAR OR E(X) don’t lack
Always write down an expression relating two equations, one known and one you got
Now use e(x) and var(x) trandfkmstjon ruled to find it out!
Some post clarity on the uniform distribution
Let n be the total numbers in the list,
This is n-first. Number +1 = Nth term, and solve for n
If nth term is n, then it’s calm.
HOWEVER DONT LACK, do NTH TERM YH but starting from the first number, so like 11, rather than 1, so work out sequence in between, not just from 1 (you want the numbers in between so work it out )
- This amount of total numbers represents the probability now 1/n
- and to find E(x) car (x) plug THIS in for n no problem, but only after you de done your linesr transformation or whatever
Not that bad just remember how to find total numbers
= N- a + 1 = nth term, and rearrange
-
Rob ability is 1/ total numbers
And can put this in for e (x) etc, after you’ve done linesr transformation
ONE MORE CRUCIAL POINT
For discrete
When working with inequalities, always multiply through and take the WHOLE NUMBER, this is bevaude itsDISCRETE, so can’t have 3/8
Discard this and go to K, now use your method and it works perfectly
DOMT FORGET
