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Lecture 8: Random Variables and Their Distributions | Statistics 110

Harvard University・41 minutes read

The text discusses the key concepts of binomial distribution, including parameters, probability mass function, and the relationship between binomial and Bernoulli distributions. It also delves into the differences between binomial and hypergeometric distributions, emphasizing factors like independence and sampling with replacement.

Insights

  • Binomial distribution in statistics represents the number of successes in independent trials, with different N and P values creating a family of distributions.
  • The distinction between binomial and hypergeometric distributions lies in independence: binomial assumes independence, while hypergeometric involves sampling without replacement, affecting the probability of subsequent events.

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Recent questions

  • What is the binomial distribution?

    The binomial distribution represents the number of successes in a fixed number of independent trials, with each trial having the same probability of success.

  • What is the probability mass function (PMF)?

    The probability mass function (PMF) for a random variable specifies the probabilities of it taking on different values, ensuring the sum of all probabilities equals 1.

  • How are discrete and continuous random variables different?

    Discrete random variables take on specific values, while continuous random variables can take any real number within a range.

  • How is the sum of two binomials calculated?

    The sum of two binomial distributions is another binomial distribution, achieved through the independence and identical distribution of the variables.

  • What distinguishes the hypergeometric distribution from the binomial distribution?

    The hypergeometric distribution involves sampling without replacement, unlike the binomial distribution, which assumes independent trials with replacement.

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Summary

00:00

Understanding Binomial Distribution in Statistics

  • Binomial distribution is a key concept in statistics, denoted as bin of NP, with parameters N and P.
  • There is a whole family of binomial distributions based on different values of N and P.
  • The distribution represents the number of successes in N independent trials, where success is defined based on the situation.
  • Binomial distribution can also be understood through indicator random variables, where XJ is 1 for success and 0 for failure in each trial.
  • The concept of iid (independent and identically distributed) is crucial in understanding the independence and distribution of random variables.
  • The probability mass function (PMF) for binomial distribution is n choose K * P^K * Q^(N-K), where Q = 1 - P.
  • A random variable assigns a numerical value to each possible outcome in the sample space.
  • The cumulative distribution function (CDF) describes the probabilities of events related to the random variable.
  • Discrete random variables take on specific values, while continuous random variables can take any real number.
  • The PMF for a discrete random variable specifies the probabilities of it taking on different values, satisfying the conditions of being greater than or equal to 0.

16:50

"Sum of PMF equals 1 for probabilities"

  • The sum over all J of PJ equals 1 in a probability mass function (PMF) to ensure the total probability is 1.
  • For a valid PMF, the sum must equal 1, indicating a complete list of possibilities.
  • PMF conditions include the sum equalling one and listing all possible values.
  • PMFs are preferred over cumulative distribution functions (CDFs) for discrete random variables.
  • The binomial distribution's PMF is easier to use than the CDF for discrete cases.
  • The binomial PMF must add up to 1, confirmed through the binomial theorem.
  • The sum of two binomials is binomial n plus M P due to independence and identical distribution.
  • Adding two functions in probability involves computing both functions and summing their values.
  • The sum of independent Bernoulli random variables results in a binomial distribution.
  • The PMF of the sum of two random variables can be computed using the law of total probability and the Vandermonde identity.

33:48

"Hypergeometric Distribution: Aces in a Deck"

  • The problem involves finding the PMF for the number of aces in a deck of cards, which can be 0, 1, 2, 3, or 4.
  • The distribution is not binomial due to the lack of independence between trials, as drawing aces affects the probability of drawing more aces.
  • The PMF is calculated by considering the probability of selecting a certain number of aces out of the total number of cards in a hand.
  • The problem is akin to the elk problem, where tagged elk are compared to tagged cards (aces) in the deck.
  • The distribution is named hypergeometric and is defined by the story of selecting marbles from a jar with black and white marbles.
  • Sampling without replacement distinguishes the hypergeometric distribution from the binomial distribution.
  • The hypergeometric distribution's PMF is validated by ensuring non-negativity and summing to one, using the Vandermonde identity.
  • The CDF is discussed, with a visual representation provided for both continuous and discrete cases.
  • A continuous CDF increases towards one as X increases and approaches zero as X decreases.
  • A discrete CDF exhibits jumps at each possible value of X, with open circles indicating the higher value at each jump.
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