Discrete Variables (bar Graph)

Discrete Variables (bar graph)

[pic 1]

Md – middle value (position = )[pic 2]

Continuous Variables (histogram)

• Each class contains its lower limit but not its upper
• Area (f or f* ) = Width (X) . Height (density)
• P ( X = a) = 0
•  = f*1X1 +…+f*n(Xn)[pic 3]
• Md and percentiles follow:

1. From F* find class which contains data
2. how much has been covered up till start of class (a)
3. how much to be covered to reach the md
4. A= wh i.e. (Md – a)h = A

Density curves

• Bell - symmetric
• Right tail Dist. – majority lower values  > md[pic 4]
• Left tail Dist. – majority higher values  < md[pic 5]

Range = max – min

Interquartile range = C75 – C25 = Q3 – Q1

V(x) =  -  **note =  = f*( i.e. the X value gets squared![pic 6][pic 7][pic 8][pic 9]

S(x) = [pic 10]

Standard Score = Z = [pic 11]

Normal Distribution  [pic 12]

 -  or md – bell’s center[pic 13][pic 14]

 – S(x) – width of bell[pic 15]

standard normal distribution (z – table) [pic 16]

• P(z(a)[pic 17]
• P(z>a) = 1 - (a)[pic 18]
• (-a) = 1-(a)[pic 19][pic 20]
• P(a(b) - (a)[pic 21][pic 22]
• (a) = 0.8        (or any number)[pic 23]

• (0.84) = 0.7995 (one below)[pic 24]
• (0.85) = 0.8023 (one above)[pic 25]
• [pic 26]

• (a) = 0.33 (any number below 0.5)[pic 27]

• (a) = 1 – 0.33[pic 28]
• (-a) = 0.67[pic 29]

Probability

AUB – union – all values within A and B

AB – intersection – only values included in both[pic 30]

AB =  – disjoint or mutually exclusive events [pic 31][pic 32]

  repeat N times count how many times A happens n(A)[pic 33]

notes:

1) 0   3) P(   4) P() = 1- P(A)   5) if AcB P(A)[pic 34][pic 35][pic 36]

For any Event!!:

P(AUB) = P(A) + P(B) – P(AB)[pic 37]

If  is symmetric then P(w) =  and P(A) = [pic 38][pic 39][pic 40]

P(A/B) =   and P(AB) = P(A) X P(B/A) = P(B) X P(A/B)[pic 41][pic 42]

Independent Events

1. P(E/F) = P(E)
2. P(F/E) = P(F)
3. Theorem if E & F are Independent then P(EF) = P(E) X P(F)[pic 43]
4. 3 events – P(ABC) = P(A) X P(B) X P(C)[pic 44][pic 45]

If A & B are disjoint i.e. AB =  then P(A/B) = 0 and not P(A) thus they are necessarily dependent.[pic 46][pic 47]

RV’s

E(x) = x1p1+…+xnpn

• if X = a and P(a) = 1 then E(x) = a
• y = ax then E(y) = E(ax) = aE(X)
• y=x+b then E(y) = E(x) + b
• E(w) = E(X+Y) = E(X) + E(Y)

V(x) = E(X2) – (E(X))2

• V(a) = 0
• y = ax then V(y) = V(ax) = a2v(X)
• y=x+b then V(y) = V(x)
• V(w) = V(X+Y) = V(X) + V(Y) +2cov(X,y)

Binomial  - finds the probability that K successes will occur in n number of attempts [pic 48]

P(x=k) = [pic 49]

And  = [pic 50][pic 51]

E(x) = n.p

V(x) = n.p.q where q = (1-p)

At least K

P(XK) = 1- P(X[pic 52]

P( at least K) = 1 – P ( not k)

For example

P(X2) = 1- P(X=0) – P(X=1)[pic 53]

P(X1) = 1- P(X<1) = 1 – P(X=0) (if its 2 people square it etc.) [pic 54]

If  P(X=0) = (1-P)^n

Geometric  - finds the probability that a success will occur for the 1st time on the nth attempt.[pic 55]

P(x=k) = [pic 56]

1. P(x>K) = [pic 57]
2. P(Xk)
3. (x=L/x>K) = P(X= L-K)

E(X) = [pic 58]

V(X) =  if p = 0 then V(X) = [pic 59][pic 60]

Correlation

Cov(X;Y) = E(XY) – E(X).E(Y)

[pic 61]

notes:

• if X= ax +b and Y= cy +d then cov(XY) = acCov(XY) but [pic 62]
• -1[pic 63]
• if X and Y are dependent they are not necessarily correlated they may be dependent in a non linear way but if correlated then they are dependent
•   iff y = ax + b[pic 64]
• [pic 65]
• if Cov(XY) >0 then V(X+Y) inc
• if Cov(XY) <0 then V(X+Y) dec