1. 1. Probability
1.1. Probability Model
1.1.1. Sample Space S: All possible outcomes
1.1.2. Event E: One outcome
1.1.3. Finds probability of E happening in S
1.2. Important Probability Results
1.2.1. 0 ≤ P(E) ≤ 1
1.2.2. P(E) = 1 → E will happen
1.2.3. P(E) = 0 → E will never happen
1.2.4. Mutually Exclusive
1.2.4.1. P(A∩B) = 0
1.2.4.2. P(A∪B) = P(A) + P(B)
1.2.5. Independent Events
1.2.5.1. One event's result does not affect chances of other events
1.2.5.2. P(A∩B) = P(A) × P(B)
1.3. Probability Tree
1.3.1. Link each outcome to the next outcome
1.4. Conditional Probability
1.4.1. If B has occurred, what is the probability of A?
1.4.2. If mutually exclusive, P(A | B) = 0
1.4.3. If independent, P(A | B) = P(A)
2. 3. Continuous Random Variable
2.1. Takes all values in an interval [a,b]
2.1.1. Described by a Probability Density Function (PDF) f(x)
2.1.2. For all Continuous Probability Distributions, probability of any individual outcome is zero
2.2. Cumulative Distribution Function (CDF) of X
2.2.1. f is the PDF of the continuous variable X
2.2.2. If a ≤ b, then F(a) ≤ F(b)
2.2.3. P(a ≤ X ≤ b) = F(b) - F(a)
2.3. Normal/Gaussian Distribution A class of continuous probability distribution
2.3.1. Probability distribution function
2.3.2. mean = μ variance = σ^2 standard deviation = σ
2.3.3. Standard Normal Distribution
2.3.3.1. Standardized Variable Z with mean = 0 and standard deviation = 1
2.3.4. Approximation of Binomial Distribution to Normal Distribution
2.3.4.1. Mean = np
2.3.4.2. Standard deviation = √np(1-p)
3. 2. Discrete Random Variable
3.1. Discrete Random Variable
3.1.1. Probabilities must be between 0 and 1
3.1.2. Sum of all probabilities = 1
3.1.3. Expected Value or Mean μ of DRV
3.1.4. Variance of DRV
3.1.5. Expected Value of f(X)
3.2. Function of a Random Variable
3.2.1. General Results
3.2.1.1. E(a) = a
3.2.1.2. E(aX) = aE(X)
3.2.1.3. E(aX) + b = aE(X) + b
3.2.1.4. E(aX) + E(bY) = aE(X) + bE(Y)
3.3. Distributions
3.3.1. Bernoulli Distribution
3.3.1.1. Probability Distribution Function Where p = P(success), q = P(failure) = 1 - p, and p + q =1
3.3.2. Binomial Distribution
3.3.2.1. Probability Distribution Function Where r is the number of successes after n trials
3.3.2.2. Mean of a Binomial Distribution
3.3.2.3. Variance of a Binomial Distribution
3.3.3. Geometric Distribution
3.3.3.1. Probability that first success occurs after n trials
3.3.3.2. Mean of a Geometric Distribution
3.3.3.3. Variance of a Geometric Distribution
4. 4. Statistics
4.1. Rappresentation
4.1.1. Histogram
4.1.2. Comulative Frequency
4.1.3. Stem-and-leaf
4.2. Measures
4.2.1. Central Tendency
4.2.1.1. Mean= (Sum of all data points)/(Number of data points)
4.2.1.2. Median
4.2.1.2.1. If the number of observations (n) is ODD: the median is the value at position. (n+1)/2
4.2.1.2.2. If the number of observations (n) is EVEN: 1. Find the value at position (n/2). 2. Find the value at position (n+1)/2. 3. Find the average of the two values to get the median.
4.2.1.3. Mode
4.2.1.3.1. Is the value that appears most often in a set of data.
4.2.2. Spread (of Variation)
4.2.2.1. Range
4.2.2.1.1. The difference between the highest and lowest numbers.
4.2.2.2. IQR (Interquartile Range) & Percentiles
4.2.2.2.1. IQR describes the middle 50% of values when ordered from lowest to highest. First find the median of the lower and upper half of the data. These values are quartile 1 (Q1) and quartile 3 (Q3). The IQR is the difference between Q3 and Q1.
4.2.2.2.2. Percentiles indicate the percentage of scores that fall below a particular value. They tell you where a score stands relative to other scores.
4.2.2.3. Variance and Standard Deviation
4.2.2.3.1. Variance is the expectation of the squared deviation of a random variable from its mean.
4.2.2.3.2. Standard Deviation is a measure of the amount of variation or dispersion of a seri of values. A low standard deviation indicates that the values tend to be close to the mean of the set, while an high standard deviation indicates that the values are spread out over a wider range.