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Chapter 5: Probability Distribution

Form 5 Additional Mathematics Bab 5: Probability Distribution

5.1 Random Variable

  • A random variable is a variable with numeric values determined from a random phenomenon, represented by X; values are represented by r. (p.142)
  • A sample space is the set of all possible outcomes of an experiment. (p.142)
  • A discrete random variable has a countable number of values (usually zero and positive integers). (p.143)
  • A continuous random variable takes values within an interval, e.g. Y = {y : a ≤ y ≤ b}. (p.144)
  • If X is a discrete random variable with probabilities P(X = ri), then Σ P(X = ri) = 1 and each P(X = ri) ≥ 0. (p.145)
  • P(A) = n(A)/n(S); 0 ≤ P(A) ≤ 1; P(A′) = 1 − P(A). (p.145)
  • A probability distribution can be shown by a tree diagram, a table, or a graph of P(X = r) against r. (p.148)
  • For independent trials (with replacement), later choices do not depend on earlier ones. (p.148)

5.2 Binomial Distribution

  • A Bernoulli trial has only two outcomes, "success" and "failure"; P(success) = p is constant, P(failure) = 1 − p; X = {0, 1}. (p.152)
  • A binomial experiment consists of n similar, independent Bernoulli trials. (p.153)
  • A binomial random variable is the number of successes r from n Bernoulli trials; its distribution is a binomial distribution. (p.153)
  • A distribution with three or more possible outcomes per trial, or with changing probabilities (without replacement), is not binomial. (p.154)
  • Binomial probability: P(X = r) = nCr pr qn−r, r = 0, 1, 2, ..., n, where q = 1 − p. (p.155)
  • This is written X ~ B(n, p). (p.155)
  • For "at least" problems, use the complement, e.g. P(X ≥ 2) = 1 − [P(X = 0) + P(X = 1)]. (p.157)
  • Probability distribution graph: when p = 0.5 it is symmetrical; p < 0.5 skewed left; p > 0.5 skewed right. (p.158)
  • Mean, μ = np. (p.161)
  • Variance, σ2 = npq. (p.162)
  • Standard deviation, σ = √(npq). (p.162)
  • To find the minimum n for P(X ≥ 1) > k, use P(X = 0) < 1 − k and take logarithms. (p.163)

5.3 Normal Distribution

  • A normal distribution can estimate a binomial distribution when n is large: np ≥ 10 and n(1 − p) ≥ 10. (p.166)
  • A normal distribution is the probability function of a continuous random variable; it is symmetrical and bell-shaped, with Mean = Median = Mode. (p.166)
  • The curve is symmetrical about X = μ, has its maximum value at X = μ, and never touches the x-axis. (p.166)
  • Total area under the graph = 1 unit2; the mean divides the area into two equal parts. (p.166)
  • Notation: X ~ N(μ, σ2). (p.166)
  • Changing μ moves the axis of symmetry left or right without changing the shape. (p.167)
  • A larger σ gives a wider, lower graph; a smaller σ gives a narrower, taller graph. (p.167)
  • The area under the graph from a to b represents P(a < X < b), which equals P(a ≤ X ≤ b). (p.167)
  • P(−∞ < X < ∞) = 1. (p.168)
  • Law of large numbers: the larger the sample size, the smaller the random variation; the experimental mean approaches the theoretical mean. (p.169)
  • A standard normal distribution has mean 0 and standard deviation 1: Z ~ N(0, 1). (p.170)
  • Standardising: Z = (X − μ)/σ. (p.170)
  • For Z: E(Z) = 0 and Var(Z) = 1. (p.171)
  • 68% of data lies within ±1 standard deviation; 95% within ±2; 99.8% within ±3. (p.171)
  • To find a z-score from X: Z = (X − μ)/σ; to find X from Z: X = μ + Zσ. (p.172)
  • P(a < X < b) = P[(a − μ)/σ < Z < (b − μ)/σ]. (p.173)
  • The standard normal table gives P(Z > a), the area of the right tail. (p.174)
  • By symmetry, P(Z > a) = P(Z < −a); P(Z > 0) = 0.5. (p.174)
  • Always sketch the standard normal graph first before reading the table. (p.175)
  • P(Z < a) = 1 − P(Z > a); P(|Z| > a) = 2P(Z > a). (p.176)
  • To find an unknown μ or σ, standardise, find the z-score from the table, then solve the equation. (p.178)
  • In problems, convert to Z, sketch the graph, and use the table; multiply the probability by the total number for expected counts. (p.181)
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