Probability - Lecture Notes PDF

# Lec 1 Random Variables - Chapter 1 ## ☑ Random Variable (متغير العشوائي) * It is a function whose domain is the sample space of a random experiment and whose range is a subset of the real numbers. It is divided into two types: ### 1. Discrete Random Variables (متغير عشوائي متقطع) * It takes disc...

# Lec 1 Random Variables - Chapter 1 ## ☑ Random Variable (متغير العشوائي) * It is a function whose domain is the sample space of a random experiment and whose range is a subset of the real numbers. It is divided into two types: ### 1. Discrete Random Variables (متغير عشوائي متقطع) * It takes discrete values that are countable, which can be finite or infinite. * For example, the number of cars passing a specific location for a limited time, or the number of heads appearing when flipping a fair coin three times. * The values it can take are 1, 2, and 3, like this: X= 1, 2, 3. ### 2. Continous Random Variables (متغير عشوائی متصل) * It takes values over an interval, which can be finite or infinite. * For example, the lifetime of a lamp or the waiting time for a certain event. * The values it can take are: * ... 1, 2, 3, 4 ... ## ☑ Discrete Random Variables (المتغير العشوائي المتقطع) * For every discrete random variable, the function F(x) equals the probability of the random variable X taking the value x. * The value of the random variable is exactly one. * It is symbolized by F (X) which is a function that takes a value of x, and gives the value of P(X=x). ## ☑ Probability Mass Function (داله الكتلة الاحتمالية) or Probability Distribution Function (داله التوزيع الاحتمالي) * F (X) = P (X=x) * It has the following conditions: * F(xi) ≥ 0 * ∑ F (Xi) = 1 for i = 1, 2, 3,.... * It is defined in terms of the values of the random variable and the probabilities of those values. * F(xi) is the probability that the random variable X takes the value xi. ## ☑ Cumulative Distribution Function (واله التوزيع التراكمية) * It is the probability that the random variable takes a value less than or equal to x. ## ☑ Expectation value (التوقع الرياضي) * If F(xi) is the probability mass function of the random variable X, then the expected value of X, denoted by E(X), is the weighted average of the values taken by X. * E(X) = Σ Xi F(xi) ### 1. Properties of Expectation * E (c) = C (The expectation of a constant is the same as the constant itself). * E (ax + b) = a E(X) + b * Var (X) = E(X2) - (E(X))2 = σ2 * σ=√var(x) * E (x2) = Σ Xi2 F(xi) ### 2. Properties of Var * Var (c) = 0 * Var (ax) = a2 var (X) * Var (ax + b) = a² Var(X) ## EX2 - given the table of values for the random variable X, calculate the expectation, variance, and standard deviation of 2X-3. * E(X) = Σ Xi F(x) = 2/3 * E(X²) = Σ Xi² F(x) = 3 * Var(x) = E(X²) – (E(X))² = 3 - (2/3)² = 7/9 * σ = √Var = √(7/9) = 1.23   E(2X-3) = 2 E(X) - 3 = 2* 2/3 - 3 = 0   Var (2x-3) = 4 Var(x)= 4 * 7/9 = 28/9 ## EX3 - Given the table of values for the random variable X, Find the value of a: * X | 0 | 1 | 2 | 3 * F(X) | a | 1/9 | 2/9 | 2/9 * Since the sum of probabilities is 1: * a + 1/9 + 2/9 + 2/9 = 1 * Therefore a = 4/9 Calculate the probability, expected value, variance, and standard deviation