Skip to main content
Back

Discrete Probability Distributions and Binomial Distribution

Control buttons has been changed to "navigation" mode.
1/20
  • What is a random variable?

    A random variable is a quantitative variable whose values are determined by chance, linking the sample space to numerical values.

  • Difference between discrete and continuous random variables?

    Discrete random variables take finite or countable values (usually integers). Continuous random variables have infinitely many values.

  • What is a probability distribution?

    A probability distribution describes the probability for each value of a random variable, often shown as a table, formula, or graph.

  • What is a probability mass function (p.m.f.)?

    A p.m.f. is a function that gives the probability associated with each value of a discrete random variable.

  • State the three requirements for a valid discrete probability distribution.
    1. Random variable x with associated probabilities.
    2. Each probability satisfies 0 ≤ P(x) ≤ 1.
    3. The sum of all probabilities equals 1.
  • How to find P(X ≥ 2) given a probability distribution?

    Sum the probabilities for all values of X greater than or equal to 2: \(P(X \geq 2) = P(X=2) + P(X=3) + \cdots\).

  • What is the expected value (mean) of a probability distribution?

    The expected value is the mean of the outcomes: \(\mu = E[X] = \sum x \cdot P(x)\).

  • How to calculate variance and standard deviation of a probability distribution?

    Variance: \(\sigma^2 = E[X^2] - (E[X])^2\). Standard deviation: \(\sigma = \sqrt{\sigma^2}\).

  • What defines a Bernoulli trial?
    1. Two possible outcomes: success or failure.
    2. Trials are independent.
    3. Constant probability of success p.
  • What is a binomial random variable?

    A binomial random variable counts the number of successes in n independent Bernoulli trials with success probability p.

  • Write the binomial probability formula.

    \(P(x) = \binom{n}{x} p^x (1-p)^{n-x}\), where \(\binom{n}{x} = \frac{n!}{x!(n-x)!}\).

  • What is the mean and standard deviation of a binomial distribution?

    Mean: \(\mu = np\). Standard deviation: \(\sigma = \sqrt{np(1-p)}\).

  • How to determine if a number of successes is significantly high or low in binomial distribution?

    Use probabilities: significantly high if \(P(X \geq x) \leq 0.05\), significantly low if \(P(X \leq x) \leq 0.05\).

  • Explain the range rule of thumb for significance in probability distributions.

    Significant values lie outside \([\mu - 2\sigma, \mu + 2\sigma]\).

  • What is the probability of exactly 3 successes in 8 trials with p=0.45?

    Using binomial distribution: \(P(X=3) = 0.2568\).

  • How to calculate probability of at most 2 successes in 8 trials with p=0.45?

    \(P(X \leq 2) = 0.2201\) by summing probabilities for X=0,1,2.

  • What is the probability that fewer than 2 successes occur in 8 trials with p=0.45?

    \(P(X < 2) = P(X=0) + P(X=1) = 0.0632\).

  • How to use StatCrunch to calculate binomial probabilities?
    1. Open StatCrunch, select Stat > Calculators > Binomial.
    2. Enter n, p, and inequality.
    3. Click Compute to get probability.
  • What is the probability histogram?

    A probability histogram shows probabilities on the vertical axis instead of relative frequencies.

  • What is the difference between a valid and invalid probability distribution?

    A valid distribution has probabilities between 0 and 1 and sums to 1; invalid distributions violate these rules.