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65 Cards in this Set

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Descriptive Statistics:
Describe the properties of a large data set.
Inferential Statistics:
Use a sample from a population to make probabilistic statements about the characteristics of a population.
Population:
A complete set of outcomes.
Samples:
A subset of outcomes drawn from a population.
NOIR:
Nominal, Ordinal, Interval, Ratio.
Nominal:
Only names (bird: robin, parrot, seagull)
Ordinal:
Ordered (large, mid, small).
Interval:
Go by intervals (0-40, 41-80, 81-120) .
Ratio:
Ratios make sense, absolute zero.
Parameter:
Describe a characteristics of a population.
Sample statistic:
Describes a characteristic of a sample.
Frequency Distribution:
a table that summarizes a large data set by assigning the observations to intervals (Histogram).
Relative frequency distributions:
A table ordered by percentage.
Cumulative Frequency Distribution:
Shows the percentage of observations less than the upper bound of each interval.
Population Mean:
Mew
Samples Mean:
Xbar
Geometric Mean (Description and formula):
Used to calculate compound growth rates.
If the returns are constant over time, geometric mean equals arithmetic mean.

The greater the variability of returns over time, the more the arithmetic mean will exceed the geometric mean.

Also known as Average Annual Compound Rate of Return

Periodic R = [(1+R1)(1+R2)...(1+Rn)^(1/n)] - 1

sqroot^n of (x1 * x2 * x3 * xn)
example:
2, 3, 4
G mean = sqroot^3 (2x3x4)
= 2.88
Weighted Mean:
Places a percentage weight on sections of a problem. A mean in which different observations have different proportional influence on the mean.

X= sum of Wi x Ri = W1R1 + W2R2 + …. Whereby W is the weight (%)
Harmonic Mean:
Used to find out the average cost per shares over time.

Xharmonic = N / sum(1/Xi)

T1 = 3,000 @ $20/share
T2 = 3,000 @ $25/share
Xharmonic = 6,000/270 shares = $22.2/share
Arithmetic mean:
= (x1+x2+x3+... + xn) / n

Example:
= (2+3+4+5+6+7) / 6
Median:
It is midpoint of a population of numbers. If there is an even number then you add 2 middle numbers and divide by 2.

Advantages: resistant to outliers.
Mode:
The most common observation.

Example:
1, 2, 3, 3, 4, 4, 4, 4, 4, 5, 5, 6, 7, 8, 8, 9.
Mode is 4 because it appeared the most.

There can be more than one mode (bimodal).
Quantiles:
Like from standardized tests.

Example:
"You are in the 75th percentile."

"60% of the data points are less than the 6th percentile."
Range:
Range of the data.
Mean absolute Deviation:
Average of the absolute values of deviations from the mean.

[I(X1-mean)I + I(X2-mean)I + … + I(Xn-mean)I] / n
Standard Deviation and Variance:
How spread out the data is. The bigger the standard deviation, the bigger the deviation from the mean.

Variance = sum [(Xi – mean)^2] / n
StdDiv = sqroot (Variance)

Example:
Mean = 2
X1 = 2, X2 = 3 X4 = 5
= [(2-2)^2 + (3-2)^2 + (4-2)^2] / 3
= 1.33

Std Div = sqroot (1.33)
Sample Variance:
Same equation as the mean except you add -1 to the n.
Variance = sum of [(Xi – mean)^2] /( n-1)
Chebyshev’s Inequality:
Specifies the minimum percentage of observations that lie within k standard deviation of the mean; applies to any distribution with k>1.

Min% = 1 – (1/k^2).

Example:
The minimum percentage of obeservations of any distribtuions of 2 standard deviations of the mean?
= 1- (1/2^2) = 75%

Remember, this is for ANY distribution.
Coefficient of Variance.
Risk per unit of measure. Therefore, the lower the better.
CV = StdDiv / Mean
Sharpe Ratio:
Excess return per unit of risk. More the better.

(Rp – Rf) / StdDivP

Rf = Average risk free rate
Rp = Mean Portfolio Risk
StdDivP = Portfolio Standard deviation
Random Variable
Uncertain Number
Outcome
Realization of random Variable
Event
Set of one or more outcomes
Mutually exclusive
Cannot both happen.
Exhaustive
Set of events includes all possible outcomes.
Empirical:
Based on analysis of data.
Using past information to make a prediction about a future outcome.
Subjective:
Personal opinion.
A Priori:
Based on reasoning, not experience.
Odds for:
Example: 20% of the time a horse will win.
Odds against:
Example: 80% of the time a horse will lose.
Unconditional:
Probability of an event that is independent of other events.
P(A).
Conditional:
A probability that is dependent on other events.
P(A|B): Probability of A given B.
What are the probability rules?
Addition rule:
P (A or B) = P(A) + P(B) - P(AB).

Multiplication (Joint Probability):
P(AB) = P(A|B) x P(B)

Multiplication (Independent Pr.):
P(AB) = P(A) x P(B)
Total Probability Rule:
P(A) = P(A|B) x P(B) + P(A|Bc) x P(Bc)

P(A) = P(AIS1) x P(S1) + P(AIS2) x P(S2) + ...
Give example of a joint probability:
P(Interest rate increased) =
P(I) = 0.4
P(Recession given at a rate increase) = P(R|I)

The probability of a recession and an increase in rates =
P(R|I) x P(I) = 0.7 x 0.4 = 28%
Give equation for joint probability events and explain it.
How is this different from mutually exclusive?
P(A or B) = P(A) + P(B) - P(A|B).

You must subtract P(A|B) because if you look at a Venn diagram you will notice that you have counted the overlapping section of two circles more than once.
Mutually exclusive events do not require the final subtraction.
Dependent Events:
Knowing the outcome of one tells you something about the probability of the other event.
Independent Events:
Occurrence of one event does not influence the occurrence of the other.
Covariance:
A measure of how two variables move together.
The value range from + infinity, to - infinity.

∑[(x-mean) / (Sx)]
Correlation:
p(Ri, Rj) = (Cov(Ri,Rj) / σ(Ri)σ(Rj)

Small p represents correlation.
Expected Value:
∑P*x

Example:
75% chance that you will receive $125,000.
The expected value is 93,750.
Portfolio Expected Value:
Weighted Average basically.
Portfolio Variance and Standard Deviation.
Var(Rp) = σ²₁w²₁+ σ₂²w₂² + 2w₁w₂p₁₂σ₁σ₂

note: p is correlation. Derived from correlation formula.
Factorial:
Example:
3! = 3 x 2 x 1
6! = 6 x 5 x 4 x 3 x 2 x1
When should you use nCr and when should use nPr?
nCr when order does not matter.
nPr when order does matter.
Sign for sample mean?
Sign for population mean?
Sample mean = x bar
Population mean = μ
Explain the geometric mean formula:
Simply multiply all the numbers given; X1 x X2 x Xn... then take the root to the power of however many observations there are (n).

Since you can not take the root of a negative number you must instead add each return by 1 then multiply;
(X1+1) x (X2+1) x (Xn+1)...
then take the root of that to the power of n.
What is MAD and how is it different from variance?
MAD uses an absolute and is not squared.

MAD =
[|X1-mean| + |X2-mean| + |Xn-mean|] / n

Variance =
[(X1-mean)^2 + (X2-mean)^2 + (Xn-mean)^2] / n

So, basically instead of absolutes you have squares.
Standard deviation =
Square root of the variance.
What's the difference between the sample variance formula and the variance formula?
the -1 at the divisor.
Coefficient of Variation?
StDiv / Mean.
Not bloody hard.
Sharpe Ratio?
The sharpe ratio is simply the mean return from an investment minus a risk free rate (like a t-bill) all divided by the standard deviation of that investment.

The investment can be a portfolio.
Kurtosis?
Leptokurtic?
Platykurtic?
Mesokurtic?
Kurtosis is a measure of the peek like q in a frequency filter.

Leptokurtic = more peek.
Platykurtic = less peek.
Mesokurtic = normal peek.
Sample Skewness?
How skewed the sample is.
(1/n)[(X - μ)^3 / (σ^3)]
Sample Kurtosis?
Same as sample skewness but replace the 3s with 4s.