# Discrete distributions_bern_binom_poisson_geom_imported Flashcards

1
Q

What is the Bernoulli distribution?

A

The Bernoulli distribution models a random experiment with two outcomes: success (1) and failure (0).

2
Q

How is the Bernoulli distribution defined mathematically?

A

The PMF of the Bernoulli distribution is: P(X = x) = p^x * (1 - p)^(1 - x), where x is 0 or 1, and p is the probability of success.

3
Q

What are the mean and variance of the Bernoulli distribution?

A

Mean: E(X) = p, Variance: Var(X) = p * (1 - p), where x is 0 or 1, and p is the probability of success.

4
Q

Can you provide an example of a Bernoulli distribution using a fair coin flip?

A

Example: Fair coin flip - P(X = 1) = 0.5, P(X = 0) = 0.5

5
Q

How about an example with a biased coin?

A

Example: Biased coin - P(X = 1) = 0.3, P(X = 0) = 0.7

6
Q

Can you provide a real-world application example of the Bernoulli distribution?

A

Example: Online ad CTR - P(X = 1) = 0.1, P(X = 0) = 0.9 (10% chance of clicking, 90% chance of not clicking).

7
Q

What is the Probability Mass Function (PMF) for the Bernoulli distribution?

A

The PMF of the Bernoulli distribution is given by: P(X = x) = p^x * (1 - p)^(1 - x), where x is the value (usually 0 or 1) and p is the probability of success.

8
Q

Could you provide an example of PMF for a biased coin with p = 0.7?

A

For a biased coin with probability of heads p = 0.7, the PMF is: P(X = 1) = 0.7, P(X = 0) = 0.3.

9
Q

What is the Cumulative Distribution Function (CDF) for the Bernoulli distribution?

A

The CDF of the Bernoulli distribution is: CDF(X ≤ x) = 1 - (1 - p)^x, where x is the value and p is the probability of success.

10
Q

Can you provide an example of CDF for the biased coin with p = 0.7?

A

For the biased coin with p = 0.7, the CDF for X ≤ 1 is calculated as: CDF(X ≤ 1) = 0.7.

11
Q

How is the Expectation (Mean) calculated for the Bernoulli distribution?

A

The expectation (mean) of a Bernoulli-distributed random variable is given by: E(X) = p, where p is the probability of success.

12
Q

What’s the expectation for the biased coin with p = 0.7?

A

For the biased coin with p = 0.7, the expectation is: E(X) = 0.7.

13
Q

What’s the formula for Variance in the Bernoulli distribution?

A

The variance of a Bernoulli-distributed random variable is: Var(X) = p * (1 - p), where p is the probability of success.

14
Q

How about an example of Variance for the biased coin with p = 0.7?

A

For the biased coin with p = 0.7, the variance is calculated as: Var(X) = 0.7 * (1 - 0.7) = 0.21.

15
Q

What is the expectation in probability and statistics?

A

The expectation, often referred to as the “expected value” or “mean,” is a fundamental concept in probability and statistics. It represents the average or central value of a random variable, considering the probabilities associated with each possible outcome.

16
Q

How is the expectation calculated for a discrete random variable?

A

For a discrete random variable X with values x₁, x₂, …, xₙ and corresponding probabilities p(x₁), p(x₂), …, p(xₙ), the expectation E(X) is calculated by multiplying each value by its probability and summing the results: E(X) = x₁ * p(x₁) + x₂ * p(x₂) + … + xₙ * p(xₙ)

17
Q

Can you provide an example of calculating expectation?

A

Certainly. For a fair six-sided die roll with equal probabilities, the expectation of the outcome (X) is calculated as: E(X) = 1 * (1/6) + 2 * (1/6) + 3 * (1/6) + 4 * (1/6) + 5 * (1/6) + 6 * (1/6) = 3.5

18
Q

How does the expectation relate to the concept of an average?

A

The expectation is akin to an average, but it considers not only the values themselves but also the likelihood of each value occurring. In essence, the expectation is a weighted average, where the weights are the probabilities of each outcome.

19
Q

What is the significance of the expectation in probability and statistics?

A

The expectation provides valuable information about the center or typical value of a random variable’s distribution. It helps in making informed decisions and predictions based on probabilistic outcomes. Understanding the expectation is crucial for assessing risks, estimating future values, and designing strategies.

20
Q

What is variance in probability and statistics?

A

Variance is a statistical measure that quantifies the spread or dispersion of data points or the variability of a random variable. It indicates how much individual data points deviate from the mean or expected value.

21
Q

How is variance calculated for a set of data?

A

For a set of data points x₁, x₂, …, xₙ with mean µ, the variance is calculated as the average of the squared differences between each data point and the mean.

22
Q

What is the formula for calculating variance of a discrete random variable?

A

For a discrete random variable X with values x₁, x₂, …, xₙ and corresponding probabilities p(x₁), p(x₂), …, p(xₙ), the variance Var(X) is calculated using the squared differences between each value and the expectation, weighted by their probabilities.

23
Q

Can you provide an example of calculating variance for a set of data?

A

Certainly. For the data set {3, 5, 7, 10, 12} with a mean of 7.4, the variance is calculated as approximately 10.96.

24
Q

How does variance relate to the concept of data spread?

A

Variance quantifies how much data points spread out around the mean. A larger variance indicates greater spread or variability, while smaller variance indicates less spread.

25
Q

What is the significance of variance in statistical analysis?

A

Variance is crucial for comparing data spread, assessing reliability, making predictions, and understanding outcome distribution. It is a key element in statistical methods like hypothesis testing, regression, and quality control.

26
Q

Is the expectation related to the population or sample in statistics?

A

The concept of expectation is related to both population and sample in statistics. It is referred to as the “population mean” or “population expected value” when considering an entire population, and as the “sample mean” when dealing with a subset of data from a larger population.

27
Q

What is the population expectation (population mean)?

A

In the context of a population, the expectation is commonly referred to as the “population mean” or “population expected value.” It represents the average value of the entire population, considering the probabilities or frequencies of each possible value.

28
Q

What is the sample expectation (sample mean)?

A

When dealing with a sample from a larger population, the expectation is referred to as the “sample mean.” It represents the average value of the observed data points in the sample and is often used as an estimate of the population mean.

29
Q

Can you provide an example of population and sample expectation?

A

Certainly. Imagine a population of exam scores. The population expectation (population mean) represents the average score for the entire population. If you take a random sample of exam scores from this population, the sample expectation (sample mean) represents the average score within the sample and estimates the population mean.

30
Q

How does the expectation concept apply to both populations and samples?

A

The concept of expectation provides insights into the central tendency of data. It represents the “center” or typical value, considering probabilities or frequencies. In populations, it’s the “population mean,” while in samples, it’s the “sample mean” used to estimate the population mean.

31
Q

What specific terms distinguish between population and sample expectations?

A

In populations, the expectation is called the “population mean” or “population expected value.” In samples, it’s referred to as the “sample mean.” These terms differentiate between the context of an entire population and a subset of data observed in a sample.

32
Q

What is the Binomial distribution?

A

The Binomial distribution is a discrete probability distribution that describes the number of successes in a fixed number of independent Bernoulli trials.

33
Q

What is the formula for the Binomial distribution?

A

The formula for the Binomial distribution is: P(X=k)=(kn​)⋅pk⋅(1−p)n−k, where n, k, p, and (kn​) are defined.

34
Q

Could you provide an example illustrating the Binomial distribution?

A

Certainly! In a spam email detection project, with 1000 emails and a 10% spam probability, the probability of having exactly 3 spam emails is calculated using the Binomial distribution formula.

35
Q

What is the Cumulative Distribution Function (CDF) for the Binomial distribution?

A

The CDF for the Binomial distribution provides the probability that the random variable X takes on a value less than or equal to x, helping to understand cumulative probabilities of different outcomes.

36
Q

Could you provide the formula for the CDF of the Binomial distribution?

A

Certainly! The formula for the CDF of the Binomial distribution is F(x)=P(X≤x)=∑k=0x​(kn​)⋅pk⋅(1−p)n−k where x, n, p, and (kn​) are defined.

37
Q

Could you illustrate the CDF of the Binomial distribution with an example?

A

Certainly! In a spam email detection project with 1000 emails and a 10% spam probability, we calculate the cumulative probability that there are 3 or fewer spam emails using the CDF formula.

38
Q

Why is understanding the CDF of the Binomial distribution important in data engineering and machine learning?

A

Understanding the CDF is essential for data engineers and machine learning practitioners. It helps analyze how probabilities accumulate from the smallest possible value to a specific x, aiding decision-making and risk assessment in various applications such as A/B testing and classification tasks.

39
Q

What is the Probability Mass Function (PMF) of the Binomial distribution?

A

The PMF of the Binomial distribution provides the probability that a Binomial-distributed random variable X takes on a specific value k, representing the number of successful outcomes in a fixed number of trials.

40
Q

Could you provide the formula for the PMF of the Binomial distribution?

A

Certainly! The formula for the PMF of the Binomial distribution is P(X=k)=(kn​)⋅pk⋅(1−p)n−k where n, k, p, and (kn​) are defined.

41
Q

How is the PMF of the Binomial distribution practically applied?

A

The PMF of the Binomial distribution is used to compute the probability of observing a specific number of successes in a fixed number of trials. For instance, it can be used in quality control to estimate the likelihood of a certain number of defects in a batch, given a known defect rate.

42
Q

What is the Multinomial distribution?

A

The Multinomial distribution is an extension of the Binomial distribution that models the probability of observing a specific combination of outcomes in a series of independent trials with multiple categories or outcomes.

43
Q

Could you provide the formula for the Multinomial distribution?

A

Certainly! The formula for the Multinomial distribution’s probability mass function (PMF) is P(X1​=x1​,X2​=x2​,…,Xk​=xk​)=x1​!⋅x2​!⋅…⋅xk​!n!​⋅p1x1​​⋅p2x2​​⋅…⋅pkxk​​, where n, k, xi​, and pi​ are defined.

44
Q

Could you provide an example illustrating the Multinomial distribution?

A

Certainly! Let’s consider rolling a 6-sided die with three colors: red, blue, and green. Rolling it 10 times, we want to calculate the probability of obtaining exactly 3 reds, 4 blues, and 3 greens using the Multinomial distribution formula.

45
Q

Why is understanding the Multinomial distribution important in data engineering and machine learning?

A

The Multinomial distribution is essential when dealing with scenarios involving multiple categories or outcomes. It’s useful for various applications, such as analyzing survey responses, text categorization, and genetic studies. Understanding it allows data engineers and machine learning practitioners to model and analyze diverse outcomes in their projects.

46
Q

What is the Probability Mass Function (PMF) of the Multinomial distribution?

A

The PMF of the Multinomial distribution provides the probability of observing a specific combination of outcomes in a series of independent trials with multiple categories or outcomes.

47
Q

Could you provide the formula for the PMF of the Multinomial distribution?

A

Certainly! The formula for the PMF of the Multinomial distribution is P(X1​=x1​,X2​=x2​,…,Xk​=xk​)=x1​!⋅x2​!⋅…⋅xk​!n!​⋅p1x1​​⋅p2x2​​⋅…⋅pkxk​​, where n, k, xi​, and pi​ are defined.

48
Q

What is the Cumulative Distribution Function (CDF) of the Multinomial distribution?

A

The CDF of the Multinomial distribution provides the probability that the random variables X1​,X2​,…,Xk​ take on values less than or equal to x1​,x2​,…,xk​, respectively.

49
Q

Could you provide the formula for the CDF of the Multinomial distribution?

A

The CDF of the Multinomial distribution involves summing up the PMF values for all possible combinations of outcomes that meet the specified conditions. However, explicit formulas for the CDF are usually not provided due to complexity. Software tools or libraries can help compute the CDF numerically. Understanding the PMF and CDF of the Multinomial distribution is essential for modeling and analyzing scenarios with multiple outcomes.

50
Q

What is the Poisson distribution?

A

The Poisson distribution is a probability distribution that describes the number of events occurring in a fixed interval of time or space, given a known average rate of occurrence, and assuming rare and independent events.

51
Q

What is the formula for the Poisson distribution?

A

The probability mass function (PMF) of the Poisson distribution is given by the formula: P(X = k) = (λ^k * e^(-λ)) / k! Where: P(X = k) is the probability of observing k events in the interval. λ (lambda) is the average rate of occurrence of events. e is the base of the natural logarithm (approximately 2.71828). k is the actual number of events observed.

52
Q

Can you provide an example of the Poisson distribution?

A

Certainly! Let’s consider a call center where the average rate of incoming calls is 4 per hour. What’s the probability of receiving exactly 3 calls in the next hour? Using the Poisson formula: λ = 4 (average rate of calls per hour) k = 3 (desired number of calls) P(X = 3) = (4^3 * e^(-4)) / 3! ≈ 0.19537

53
Q

Could you give another example of the Poisson distribution?

A

Absolutely! Imagine a specific intersection where accidents occur on average 2 times per day. What’s the probability of having no accidents in a given day? Using the Poisson formula: λ = 2 (average rate of accidents per day) k = 0 (no accidents) P(X = 0) = (2^0 * e^(-2)) / 0! ≈ 0.13534

54
Q

What is the Probability Mass Function (PMF) of the Poisson distribution?

A

The PMF of the Poisson distribution gives the probability of a discrete random variable taking on a specific value. For the Poisson distribution, the PMF is given by the formula: P(X = k) = (λ^k * e^(-λ)) / k! Where: P(X = k) is the probability of observing k events in the interval. λ (lambda) is the average rate of occurrence of events. e is the base of the natural logarithm (approximately 2.71828). k is the actual number of events observed.

55
Q

What is the Cumulative Distribution Function (CDF) of the Poisson distribution?

A

The CDF of the Poisson distribution gives the probability that a random variable is less than or equal to a specific value. It’s the sum of the PMF values for all smaller values of the random variable. The CDF is given by the formula: F(X ≤ k) = Σ (λ^i * e^(-λ)) / i! Where: F(X ≤ k) is the cumulative probability that X is less than or equal to k. λ (lambda) is the average rate of occurrence of events. e is the base of the natural logarithm (approximately 2.71828). i ranges from 0 to k.

56
Q

Could you provide an example of using the PMF of the Poisson distribution?

A

Certainly! Let’s consider a call center where the average rate of incoming calls is 5 per hour. What’s the probability of receiving exactly 3 calls in the next hour using the PMF? Using the Poisson PMF formula: λ = 5 (average rate of calls per hour) k = 3 (desired number of calls) P(X = 3) = (5^3 * e^(-5)) / 3! ≈ 0.14037

57
Q

Can you show an example of using the CDF of the Poisson distribution?

A

Certainly! Continuing with the call center scenario, what’s the cumulative probability of receiving 5 or fewer calls in the next hour using the CDF? Using the Poisson CDF formula: λ = 5 (average rate of calls per hour) k = 5 (desired number of calls) F(X ≤ 5) = Σ (5^i * e^(-5)) / i! for i = 0 to 5 ≈ 0.61596

58
Q

What is the Geometric distribution?

A

The Geometric distribution is a probability distribution that models the number of trials required for the first success in a sequence of independent Bernoulli trials, where each trial has a constant probability of success denoted by “p.”

59
Q

What is the Probability Mass Function (PMF) of the Geometric distribution?

A

The PMF of the Geometric distribution is given by the formula: P(X = k) = (1 - p)^(k - 1) * p Where: P(X = k) is the probability of the first success occurring on the kth trial. p is the probability of success on a single trial. k is the trial number.

60
Q

What is the Cumulative Distribution Function (CDF) of the Geometric distribution?

A

The CDF of the Geometric distribution is given by the formula: F(X ≤ k) = 1 - (1 - p)^k Where: F(X ≤ k) is the cumulative probability that the first success occurs on or before the kth trial. p is the probability of success on a single trial. k is the trial number.

61
Q

Could you provide an example of using the Geometric PMF?

A

Certainly! Imagine you’re flipping a fair coin, and you want to know the probability of getting heads on the first flip. Using the Geometric PMF formula: p = 0.5 (probability of getting heads on a single flip) k = 1 (first trial) P(X = 1) = (1 - 0.5)^(1 - 1) * 0.5 = 0.5

62
Q

Can you show an example of using the Geometric CDF?

A

Sure! Continuing with the coin-flipping scenario, what’s the cumulative probability of getting heads within the first 3 flips? Using the Geometric CDF formula: p = 0.5 (probability of getting heads on a single flip) k = 3 (third trial) F(X ≤ 3) = 1 - (1 - 0.5)^3 ≈ 0.875

63
Q

How do you explain the examples provided?

A

In the first example, we used the Geometric PMF formula to calculate the probability of getting heads on the first flip of a fair coin. In the second example, we used the Geometric CDF formula to calculate the cumulative probability of getting heads within the first 3 flips.

64
Q

Example: Bernoulli Distribution for Email Marketing Campaign. Number of recipients who clicked (X = 1): 50

A

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65
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66
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67
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68
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