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Understanding Discrete and Continuous Random Variables in Probability Theory

This chapter explores key concepts of discrete and continuous random variables in probability theory. It begins with discrete variables, using examples like the value from a fair die roll, highlighting uniform distribution and probability histograms. The transition to continuous variables discusses continuous distribution, introducing density curves and cumulative probabilities. Essential aspects such as interval probabilities and the Fundamental Theorem of Calculus are also examined, providing a foundational understanding of how probabilities are calculated and represented graphically.

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Understanding Discrete and Continuous Random Variables in Probability Theory

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  1. CHAPTER 4 4.1 - Discrete Models General distributions Classical: Binomial, Poisson, etc. 4.2 - Continuous Models General distributions Classical: Normal, etc.

  2. Motivation ~ Consider the following discrete random variable… Example: X = “value shown on a single random toss of a fair die (1, 2, 3, 4, 5, 6)” Xis said to be uniformly distributed over the values 1, 2, 3, 4, 5, 6. Probability Histogram P(X = x) Total Area = 1 Density X “What is the probability of rolling a 4?”

  3. Motivation ~ Consider the following discrete random variable… Example: X = “value shown on a single random toss of a fair die (1, 2, 3, 4, 5, 6)” Xis said to be uniformly distributed over the values 1, 2, 3, 4, 5, 6. Probability Histogram P(X = x) Total Area = 1 Density X “What is the probability of rolling a 4?”

  4. Motivation ~ Consider the following discrete random variable… Example: X = “value shown on a single random toss of a fair die (1, 2, 3, 4, 5, 6)” Xis said to be uniformly distributed over the values 1, 2, 3, 4, 5, 6. Cumulative distribution P(Xx)

  5. Motivation ~ Consider the following discrete random variable… Example: X = “value shown on a single random toss of a fair die (1, 2, 3, 4, 5, 6)” Xis said to be uniformly distributed over the values 1, 2, 3, 4, 5, 6. Cumulative distribution P(Xx) “staircase graph” from 0 to 1

  6. POPULATION Time intervals = 0.5 secs Example: X= “reaction time” “Pain Threshold” Experiment: Volunteers place one hand on metal plate carrying low electrical current; measure duration till hand withdrawn. Time intervals = 1.0 secs “In the limit…” Continuous random variable X we obtain a density curve Time intervals = 2.0 secs Total Area = 1 Time intervals = 5.0 secs SAMPLE Time intervals = 1.0 secs In principle, as # individuals in samples increase without bound, the class interval widths can be made arbitrarily small, i.e, the scale at which X is measured can be made arbitrarily fine, since it is continuous.

  7. Cumulative probability F(x) = P(X x) = Area under density curve up to x “In the limit…” we obtain a density curve 00 f(x) = density function • f(x)  0 • Area = 1 F(x) increases continuously from 0 to 1. x x x f(x) no longer represents the probability P(X = x), as it did for discrete variables X. In fact, the zero area “limit” argument would seem to imply P(X = x) = 0 ??? (Later…) However, we can define “interval probabilities” of the form P(a X b), using F(x).

  8. Cumulative probability F(x) = P(X x) = Area under density curve up to x “In the limit…” we obtain a density curve F(b) f(x) = density function F(b) F(a) F(a) • f(x)  0 • Area = 1 F(x) increases continuously from 0 to 1. b a b a f(x) no longer represents the probability P(X = x), as it did for discrete variables X. In fact, the zero area “limit” argument would seem to imply P(X = x) = 0 ??? (Later…) However, we can define “interval probabilities” of the form P(a X b), using F(x).

  9. Cumulative probability F(x) = P(X x) = Area under density curve up to x “In the limit…” we obtain a density curve F(b) f(x) = density function F(b) F(a) F(a) • f(x)  0 • Area = 1 F(x) increases continuously from 0 to 1. b a b a f(x) no longer represents the probability P(X = x), as it did for discrete variables X. An “interval probability” P(a X b) can be calculated as the amount of area under the curve f(x) betweena and b, or the difference P(X b)  P(X a), i.e., F(b)  F(a). (Ordinarily, finding the area under a general curve requires calculus techniques… unless the “curve” is a straight line, for instance. Examples to follow…)

  10. Cumulative probability F(x) = P(X x) = Area under density curve up to x “In the limit…” we obtain a density curve f(x) = density function • f(x)  0 • Area = 1 F(x) increases continuously from 0 to 1. Fundamental Theorem of Calculus Thus, in general, P(a X b) = = F(b)  F(a). Moreover, and .

  11. Consider the following continuous random variable… Example: X = “value shown on a single random toss of a fair die (1, 2, 3, 4, 5, 6)” Probability Histogram CumulProb P(Xx) P(X = x) Total Area = 1 Density X “What is the probability of rolling a 4?” “staircase graph” from 0 to 1

  12. Consider the following continuous random variable… Example: X = “Ages of children from 1 year old to 6 years old” Further suppose thatXis uniformly distributed over the interval [1, 6]. Probability Histogram CumulProb P(Xx) P(X = x) Total Area = 1 Density X “What is the probability of rolling a 4?” “staircase graph” from 0 to 1

  13. Consider the following continuous random variable… Example: X = “Ages of children from 1 year old to 6 years old” Further suppose thatXis uniformly distributed over the interval [1, 6]. Probability Histogram CumulProb P(Xx) P(X = x) Total Area = 1 Density X “What is the probability of rolling a 4?” “staircase graph” from 0 to 1

  14. Consider the following continuous random variable… Example: X = “Ages of children from 1 year old to 6 years old” Further suppose thatXis uniformly distributed over the interval [1, 6]. > 0  CumulProb P(Xx) Total Area = 1 Density Check? Base = 6 – 1 = 5 5  0.2 = 1 Height = 0.2 X “What is the probability of rolling a 4?” that a random child is 4 years old?” doesn’t mean….. = 0 !!!!! The probability that a continuous random variable is exactlyequal to any single value is ZERO! A single value is one point out of an infinite continuum of points on the real number line.

  15. Consider the following continuous random variable… Example: X = “Ages of children from 1 year old to 6 years old” Further suppose thatXis uniformly distributed over the interval [1, 6]. CumulProb P(Xx) Alternate way using cumulative distribution function (cdf)… Density X “What is the probability of rolling a 4?” between 4 and 5 years old?” that a random child is 4 years old?” actually means.... = (5 – 4)(0.2) = 0.2 NOTE: Since P(X = 5) = 0, no change for P(4 X 5), P(4 <X5), or P(4 <X< 5).

  16. Consider the following continuous random variable… Example: X = “Ages of children from 1 year old to 6 years old” Further suppose thatXis uniformly distributed over the interval [1, 6]. CumulProb P(Xx) Alternate way using cumulative distribution function (cdf)… Density X “What is the probability of rolling a 4?” under 5 years old? that a random child is 0.8

  17. Consider the following continuous random variable… Example: X = “Ages of children from 1 year old to 6 years old” Further suppose thatXis uniformly distributed over the interval [1, 6]. CumulProb P(Xx) Alternate way using cumulative distribution function (cdf)… Density X under 4 years old? “What is the probability of rolling a 4?” that a random child is 0.6

  18. Consider the following continuous random variable… Example: X = “Ages of children from 1 year old to 6 years old” Further suppose thatXis uniformly distributed over the interval [1, 6]. CumulProb P(Xx) Alternate way using cumulative distribution function (cdf)… Density X between 4 and 5 years old?” “What is the probability of rolling a 4?” that a random child is

  19. Consider the following continuous random variable… Example: X = “Ages of children from 1 year old to 6 years old” Further suppose thatXis uniformly distributed over the interval [1, 6]. CumulProb P(Xx) Alternate way using cumulative distribution function (cdf)… Density X between 4 and 5 years old?” “What is the probability of rolling a 4?” that a random child is = F(5) F(4) =0.8 – 0.6 = 0.2

  20. Consider the following continuous random variable… Example: X = “Ages of children from 1 year old to 6 years old” Further suppose thatXis uniformly distributed over the interval [1, 6]. Cumulative probability F(x) = P(X x) = Area under density curve up to x CumulProb P(Xx) For any x, the area under the curve is Density F(x) = 0.2 (x – 1). X x

  21. Consider the following continuous random variable… Example: X = “Ages of children from 1 year old to 6 years old” Further suppose thatXis uniformly distributed over the interval [1, 6]. Cumulative probability F(x) = P(X x) = Area under density curve up to x F(x) = 0.2 (x – 1) For any x, the area under the curve is F(x) increases continuously from 0 to 1. Density F(x) = 0.2 (x – 1). (compare with “staircase graph” for discrete case) x

  22. Consider the following continuous random variable… Example: X = “Ages of children from 1 year old to 6 years old” Further suppose thatXis uniformly distributed over the interval [1, 6]. Cumulative probability F(x) = P(X x) = Area under density curve up to x F(x) = 0.2 (x – 1) F(5) = 0.8 0.2 Density F(4) = 0.6 “What is the probability that a child is between 4 and 5?” = F(5) F(4) =0.8 – 0.6 = 0.2

  23. Consider the following continuous random variable… Example: X = “Ages of children from 1 year old to 6 years old” Further suppose thatXis uniformly distributed over the interval [1, 6]. > 0  Base  Height Area = = 1  Density

  24. Consider the following continuous random variable… Example: X = “Ages of children from 1 year old to 6 years old” Base  Height Area = Density “What is the probability that a child is under 4 years old?”

  25. Consider the following continuous random variable… Example: X = “Ages of children from 1 year old to 6 years old” Alternate method, without having to use f(x): Base  Use proportions via similar triangles. Area = = 0.36 Density h = ? 0.36 “What is the probability that a child is under 4 years old?”

  26. Consider the following continuous random variable… Example: X = “Ages of children from 1 year old to 6 years old” Density 0.64 0.36 “What is the probability that a child is under 4 years old?” “What is the probability that a child is over 4 years old?”

  27. Consider the following continuous random variable… Example: X = “Ages of children from 1 year old to 6 years old” Cumulative probability F(x) = P(X x) = Area under density curve up to x Exercise… F(x) = ???????? ? Density x “What is the probability that a child is under 4 years old?” “What is the probability that a child is under 5 years old?” “What is the probability that a child is between 4 and 5?”

  28. IMPORTANT SPECIAL CASE: “Bell Curve” Unfortunately, the cumulative area (i.e., probability) under most curves either… requires “integral calculus,” or is numerically approximated and tabulated.

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