Probability and Statistics PDF

## Probability, Statistics ### Probability of Events * $P(A) = \frac{N(A)}{N(S)}$ * N(A): Number of elements * N(S): Total number of elements **Example:** If a fair coin is tossed twice, what is the probability of getting at least one head? S = {HH, HT, TH, TT} A = {HH, HT, TH} P(A)...

## Probability, Statistics ### Probability of Events * $P(A) = \frac{N(A)}{N(S)}$ * N(A): Number of elements * N(S): Total number of elements **Example:** If a fair coin is tossed twice, what is the probability of getting at least one head? S = {HH, HT, TH, TT} A = {HH, HT, TH} P(A) = 3/4 = 0.75 ### Properties of Probability 1. 0 < P(A) < 1 2. P(A) = 1 - P(A) 3. P(A∪B) = P(A) + P(B) (if A and B are mutually exclusive) 4. P(A∪B) = P(A∩B) 5. P(A∩B) = P(A∪B) 6. P(A∩B) = P(A) - P(A∩B) **Example:** If P(A) = 1/3, P(B) = 1/2, P(A∩B) = 1/6, find: P(A∪B) 1. P(A∪B) = P(A) + P(B) - P(A∩B) * = 1/3 + 1/2 - 1/6 = 2/3 2. P(A) = 1 - P(A) * = 1 - 1/3 = 2/3 3. P(B) = 1 - P(B) * = 1 - 1/2 = 1/2 4. P(A∩B) = P(A∪B) = 1 - P(A∪B) * = 1 - 2/3 = 1/3 5. P(A∪B) = P(A∩B) = 1 - P(A∩B) * = 1 - 1/6 = 5/6 ### Conditional Probability * P(A/B) = P(A∩B)/P(B) (if P(B) ≠ 0) * P(A∩B) = P(A)⋅P(B) (if A and B are independent) **Example:** Flip a coin and then roll a die. What is the probability of observing heads on the coin and 2 or 3 on the die? P(A∩B) = P(A)⋅P(B) =(1/2)⋅(1/3) = 1/6 **Example:** Let P(A) = 0.4, P(A∪B) = 0.6. For what P(B) are A and B independent? P(A∪B) = P(A) + P(B) - P(A∩B) 0.6 = 0.4 + P(B) - P(A)⋅P(B) 0.2 = 0.6 P(B) P(B) = 0.2/0.6 = 1/3 ### Bayes Theorem * P(A) = Σ P(A∩Bi) * P(A) = Σ P(Bi)⋅P(A/Bi) * P(Bi/A) = P(A∩Bi)/P(A) **Example:** Two boxes containing marbles are placed on a table. The boxes are labeled B1 and B2. Box B1 contains 7 green, 4 white marbles. Box B2 contains 3 green and 10 yellow. The probability of selecting box B1 is 1/2 and B2 is 1/3, P(B1) = 1/2 P(B2) = 1/3. Is she selecting a green? **Solution:** 1. The probability that she will win the TV: * P(A) = ΣP(A∩Bi) = P(B1)⋅P(A|B1)+P(B2)⋅P(A|B2) * = (1/2)⋅(7/11)+(1/3)⋅(3/13) = 0.365 2. The probability that the green was selected from the first box: * P(B1/A) = P(B1)⋅P(A|B1)/P(A) * = (1/2)⋅(7/11)/(0.365) = 0.58 ### Random Variables and Distribution Functions * **Discrete:** Random variable can take only finitely many values. * **Continuous:** Random variable can take any value in a given range. * **Probability Density Function (PDF):** Gives the probability of the random variable taking a certain value. * **Cumulative Distribution Function (CDF):** Gives the probability of the random variable taking a value less than or equal to a given value. * **Distribution Function of Discrete:** Σ P(X = x) = 1 * **Distribution Function of Continuous:** ∫ P(x)dx = 1 **Example:** Consider X = {1, 2, 3}. Find the CDF if the PDF is f(x) = k(2x-1). * F(x) = ∑k(2x-1) = 1 * k(1+3+5) = 1 * k = 1/9 * ∴ F(x) = (1/9)(2x-1) * F(1) = f(1) = (1/9)(2−1) = 1/9 * F(2) = F(1) + f(2) = 1/9 + (1/9)(2-1) = 2/9 * F(3) = F(1) + F(2) + f(3) = 1/9 + 2/9 +(1/9)(4-1) = 4/9 **Example:** Find the CDF for the following function: * f(x) = cx²+x; 0 ≤ x ≤ 1 * f(x) = 0; otherwise ∫(cx²+x)dx = (c(x³/3)+x²/2)|_0^1 = 1 ⇒ c = 3/2 * F(x) = (3/2)(x³/3)+x²/2 = 3/2 * x³ + x²/2; 0 ≤ x ≤ 1 ### Moment of Random Variables (Expectation) * **E(X):** The expectation of the random variable X. * **Var(X):** The variance of the random variable X. * **Discrete:** E(X) = ∑ X⋅ P(X) * **Continuous:** E(X) = ∫ X⋅ P(X)dx * **E(ax+b) = a E(X) + b (a and b are constants)** * **Var(X) = E[(X-μ)²]** * Discrete: Var(X) = Σ(X-μ)² ⋅ P(X) * Continuous: Var(X) = ∫(X-μ)²⋅ P(X) dx * **Var(ax+b) = a² Var(X)** **Example:** Find E(X), Var(X) for the variable X: | X | 1 | 2 | 5 | |---|---|---|---| | P(X) | 1/3 | 1/3 | 1/3 | * E(x) = 1(1/3)+2(1/3)+5(1/3) = 8/3 * Var(x) = E(x²) - (E(x))² = (1/3)(1² + 2² + 5²) - (8/3)² = 2.6875 **Example:** Let P(X) = 1/2 (2 ≤ x ≤ 7) and 0 otherwise. Find **E(X)**, **E(2x-3)**, and **Var(2x-1)**. * E(X) = ∫ xf(x)dx = ∫(2 ≤ x ≤ 7) (1/2)*x dx = 4.5 * E(2x-3) = 2E(x) - 3 = 2(4.5) - 3 = 6 * Var(X) = ∫(x-μ)²f(x)dx = ∫(2 ≤ x ≤ 7) (x-4.5)²(1/2) dx = 2.08 * Var(2x-1) = 2²⋅Var(X) * Var(2x-1) = 4⋅2.08 = 8.33 ### Special Discrete Distributions * **Binomial Distribution:** * P(x) = ⁿCₓ ⋅ Pˣ ⋅ (1-P)ⁿ⁻ˣ * n = Number of trials * x = Number of successes * P = Probability of success in a single trial **Example:** What is the probability of rolling two sixes and three non-sixes in 5 independent casts of a fair die? * P(x = 2) = ⁵C₂(1/6)²(5/6)³ = 0.160751 * **Poisson Distribution:** * P(x) = (e⁻λ λˣ)/x! * λ = Mean of the distribution **Example:** A random variable X has a Poisson distribution with λ = 3. What is the probability that X is bounded by 1 and 3? * P(1 < x < 3) = P(x = 1) + P(x = 2) + P(x = 3) * P(1 < x < 3) = (e⁻³ 3¹)/1! + (e⁻³ 3²)/2! + (e⁻³ 3³)/3! * P(1 < x < 3) = (e⁻³ 3¹)/1! + (e⁻³ 3²)/2! + (e⁻³ 3³)/3! * P(1 < x < 3) = 11e⁻³/2 = 0.122 ### Special Continuous Distributions * **Normal Distribution:** * f(x) = (1/σ√2π)⋅ e⁻(x-μ)²/2σ² * μ = Mean of the distribution * σ = Standard Deviation of the distribution * **Standard Normal Distribution:** * μ = 0 * σ = 1 **Example:** Find P(X<-1.2) if X~N(0,1) * P(X < -1.2) = 0.5 - P(X ≤ 1.2) * P(X < -1.2) = 0.5 - 0.3849 * P(X < -1.2) = 0.1511 **Example:** If X~N(3, 16), what is P(4<X<8)? * P(4<X<8) = P(4-3/4 < X-3/4 < 8-3/4) = P(1/4 < X-3/4 < 5/4) * P(4<X<8) = P(X < 1.25) - P(X ≤ 1.25) * P(4<X<8) = 0.3944 - 0.0987 * P(4<X<8) = 0.2957 **Example:** Find P(-1.2 ≤ x < 1.3) if x ~ N(0,1) * P(-1.2 ≤ x < 1.3) = P(0 ≤ x < 1.3) + P(x < -1.2) * P(-1.2 ≤ x < 1.3) = 0.4032 + 0.3849 * P(-1.2 ≤ x < 1.3) = 0.7881 ### Statistics * **Data:** A set of values. * **Grouped Data:** Data organized into categories (intervals). * **Ungrouped Data:** Data that is not organized into categories. **Measures of Central Tendency** * **Mean:** The average of a set of data. * M = (Σx)/n **Example:** Find the mean of the following data: 2, 5, 6, 7, 10, 4, 3 * M = (2+5+6+7+10+4+3)/7 = 37/7 = 5.282 * **Median:** The middle value in a set of data, when the data is ordered from least to greatest. * If the number of data points is even, the median is the average of the two middle values. * If the number of data points is odd, the median is the middle value. * **Mode:** The value that appears most often in a set of data. **Measures of Dispersion ** * **Variance (σ²):** A measure of how spread out the data is. * σ² = Σ(x-μ)²/n * **Standard Deviation (σ):** The square root of the variance. * σ = √σ² * **Range (R):** The difference between the largest and smallest values in a set of data. **Example:** Find σ² and σ for these data: 1, 3, 5 if μ = 3 * σ² = (1/3)[(1-3)²+(3-3)²+(5-3)²] * σ² = (1/3)[4+0+4] = 8/3 * σ = √(8/3) = √(8/3) ≈ 1.63 **Example:** From the following data, construct a frequency distribution table and find the standard deviation: 6, 8, 8, 8, 8, 13, 16, 1, 1, 11, 17, 3, 10, 4, 4, 8, 17, 3, 12, 11, 3, 17, 9, 11, 7, 2, 17, 1, 4 * R = U - L = 17 - 1 = 16 * W = R/K = 16 / 3.322 log(35). * W = **2.6** <br> | Class | Frequency (fi) | mi | mi⋅f | (mi-M)² | (mi-M)²⋅f | |---|---|---|---|---|---| | 1-3 | 9 | 2 | 18 | 36 | 324 | | 4-6 | 5 | 5 | 25 | 9 | 45 | | 7-9 | 8 | 8 | 64 | 0 | 0 | | 10-12 | 7 | 11 | 77 | 9 | 63 | | 13-15 | 2 | 14 | 28 | 36 | 72 | | 16-18 | 4 | 17 | 68 | 81 | 324 | | **Total** | **35** | 280| **828** | | **828** | * M = Σ(mi⋅f)/Σfi = 280 / 35 = 8 * σ² =(Σ(mi-M)²⋅fi)/Σfi = 828/35 = 23.7 * σ = √23.7 = 4.8