Discrete and Continuous Random Variables

Lesson 6.1 - Discrete Random Variables

  • Learning Targets:

    • LTZ: Describe a probability distribution

    • LTE: Probability distribution

  • Definition: A discrete random variable takes a countable number of values with gaps between them.

  • Probability Distribution:

    • Must add up to 1 (ΣP(X) = 1)

    • Mean or Expected Value (denoted as Mx):

    • Formula: Mx=ΣxiP(x)Mx = \Sigma x_i P(x)

  • Variability:

    • Standard Deviation (SD): O=(x−M)2P(x)O = \sqrt{(x - M)^2 P(x)}

Lesson 6.2 - Continuous Random Variables

  • Learning Targets:

    • LTZ: Probabilities

    • LT: Random variables

  • Key Concept:

    • Continuous random variables have infinite values with no gaps.

  • Examples of Continuous Distributions:

    • Uniform Distribution

    • Normal Distribution

  • Finding Probabilities:

    • To find probabilities for continuous random variables, calculate the area under the curve.

  • Notation:

    • N(M, σ) denotes the Normal distribution with mean (M) and standard deviation (σ).

Lesson 6.3 - Transforming Random Variables

  • Learning Targets:

    • Add or subtract a constant.

  • Effects of Transformation:

    • Shape: Stays the same

    • Center: Add or subtract constant (c)

    • Variability: Stays the same (range, IQR, SD)

  • Check on Variability:

    • Variance is affected by transformations: it multiplies/divides by c.

Lesson 6.4 - Combining Random Variables

  • Learning Target:

    • LTA: Combining random variables

  • Formulas:

    • Mx+y=Mx+MyMx+y = Mx + My

    • Mx−y=Mx−MyMx-y = Mx - My

  • Standard Deviation:

    • Adding or subtracting: Not simply added; must consider independence.

  • Example:

    • OXY - YO is incomplete without associated variances.

Lesson 6.5 - Binomial Random Variables

  • Learning Target:

    • LT: Understand binomial random variables

  • Definition: A binomial random variable considers the number of successes (X) in n trials.

    • Parameters: B(n, p)

    • Characteristics:

    • Binary outcomes: success or failure

    • Independent trials

    • Fixed number of trials (n)

    • Same probability of success (p)

  • Binomial Formula:

    • The probability of k successes is given by:

    • P(x=k)=(nk)pk(1−p)(n−k)P(x=k) = {n \choose k} p^k (1-p)^{(n-k)}

    • Where:

    • n = number of trials

    • k = number of successes

    • p = probability of success

Lesson 6.6 - Parameters for Binomial Distributions

  • Learning Target:

    • LT1: Using technology for calculations

  • Mean:

    • M=npM = np

    • Interpreted as the average number of successes after many trials.

  • Standard Deviation:

    • Calculated as: SD=np(1−p)SD = \sqrt{np(1-p)}

Lesson 6.7 - Conditions for Inference

  • Learning Target:

    • LT1: Understanding binomial conditions

  • 10% Condition:

    • When taking a random sample without replacement of size n from a population of size N, we can use binomial distribution if:

    • n≤0.10Nn \leq 0.10N

  • Variance from Mean:

    • The number of successes typically varies by a standard deviation from the mean (M).

  • Large Counts Condition:

    • Can model a binomial distribution using Normal distribution if:

    • np≥10extandn(1−p)≥10np \geq 10 ext{ and } n(1-p) \geq 10

Lesson 6.8 - The Geometric Distribution

  • Learning Target:

    • LT: Understanding geometric distribution

  • Definition: A geometric distribution is focused on the number of trials until the first success.

  • Characteristics:

    • Binary outcomes: success or failure

    • Independent trials

    • Some probability of success per trial

  • Geometric Formula:

    • The probability of k failures before a success is given by:

    • P(x=k)=(1−P)kPP(x=k) = (1-P)^k P

  • Shape:

    • The geometric distribution is skewed right.

  • Mean:

    • M=1pM = \frac{1}{p}

  • Standard Deviation:

    • σ=1−pp2σ = \frac{\sqrt{1-p}}{p^2}

  • Variability: Indicates the spread of the distribution in context of failures before the first success.