Module 5

• Random variable


  A variable whose value, X, is determined by a random experiment
• Random experiment


  An experiment that can result in different outcomes, even though it is repeated in the same manner every time
• Random variable example


  • Tossing a fair coin twice, where X is the number of heads obtained
• Random variable example


  • Flipping 3 coins at the same time, where X is the number of heads
• Discrete random variable


  A random variable that has a finite or countable number of possible outcomes that can be listed
• Continuous random variable


  A random variable that has an uncountable number or possible outcomes, represented by the intervals on a number line
• Cumulative distribution function (CDF)

  A function defined for any real number x as F(x)= P(X≤x)
• Properties of CDF


  • Its value ranges from 0 to 1, inclusive of the endpoints
  • A non decreasing function
  • The graph either remains flat or it goes up, never goes down
  • Every random variable will have one and only one CDF
  • CDF is used to compute for the probability of any event that is expressed in terms of a random variable
  • The behavior of CDF random variable depends on the type of random variable (discrete or continuous)
• Discrete Probability Distributions or PMF


  A table that lists each possible value the random variable can assume, together with its probability
• Constructing a Discrete Probability Distribution

  1. Make a frequency distribution for the possible outcomes

  2. Find the sum of the frequencies

  3. Find the probability of each possible outcome by dividing its frequency by the sum of the frequencies

  4. Check that each probability is between 0 and 1 and that the sum is 1
• Mean of a discrete random variable


  μ = ΣxP(x)
• Variance of a discrete random variable

  σ^2 = Σ(x - μ)^2P(x)
• Standard deviation of a discrete random variable


  σ = √(σ^2)
• Probability density function (PDF)


  A function that is defined for any real number x and satisfies: f(x) ≥ 0 for all x, the area below the whole curve f(x) and above the x-axis is always equal to 1, P(a≤X≤b) is the area bounded by the curve f(x), the x-axis, and the lines x=a and x=b
• Cumulative distribution function (CDF) of a continuous random variable


  F(x) = ∫(-∞)^x f(x) dx
• Example of finding probability using CDF

  • P(4 < X < 6.5) = F(6.5) - F(4)
• We can express the probability of the event in terms of the CDF as: P(X ≤ a) = F(a), P(X > a) = 1 - F(a), P(a < X < b) = F(b) - F(a)