Probability - Lecture Notes PDF
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Sinai University Arish
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Summary
These lecture notes cover the fundamental concepts of probability theory, focusing on random variables and their types (discrete and continuous). They also discuss probability mass functions, cumulative distribution functions, and expectation and variance calculations.
Full Transcript
# 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