The Poisson distribution models how many times something occurs in a fixed interval of time or space when occurrences happen independently at a known average rate.
Missing Word
Poisson
The Poisson distribution counts occurrences in an interval, such as calls per hour or flaws per meter of wire. Its mean and variance are both equal to mu.
Category: Poisson Distribution
Question 2
A Bernoulli trial is one trial with only two possible outcomes, such as a coin flip landing heads or tails.
Missing Word
Bernoulli
A Bernoulli trial has two outcomes with fixed probabilities. A binomial experiment counts the successes in a set number of independent Bernoulli trials.
Category: Binomial Distribution
Question 3
In a binomial experiment, the outcome being counted is labeled a success, and its probability is written p.
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success
Success is just the outcome of interest, not necessarily a good one; a defective part can be the success. The probability of a success stays the same on every trial.
Category: Binomial Distribution
Question 4
The outcome that is not being counted in a binomial experiment is labeled a failure, and its probability is q = 1 - p.
Missing Word
failure
Each binomial trial ends in either the counted outcome or a failure. Because these are the only two outcomes, their probabilities p and q add to 1.
Category: Binomial Distribution
Question 5
In the notation X ~ B(n, p), the letter n stands for the number of trials.
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trials
The number of trials, n, is fixed before the experiment starts. Together with p, it determines the binomial distribution completely.
Category: Binomial Distribution
Question 6
Binomial experiments require that each attempt be independent, so the result of one attempt has no effect on any other.
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independent
Independence means the probability of success is not changed by earlier results. Repeated coin flips are independent; dealing cards without replacement is not.
Category: Binomial Distribution
Question 7
In a hypergeometric experiment, the picks are dependent because each item drawn is not put back before the next pick.
Missing Word
dependent
Sampling without replacement changes the makeup of the remaining population after each pick. That makes later picks depend on earlier ones, which is why the binomial model does not apply exactly.
Category: Hypergeometric Distribution
Question 8
The square of the standard deviation of a random variable is its variance.
Missing Word
variance
Variance measures spread in squared units, and the standard deviation is its square root. For a binomial distribution the variance is npq.
Category: Mean and Standard Deviation
Question 9
A binomial distribution is completely determined by its two parameters, n and p.
Missing Word
parameters
Parameters are the fixed numbers that pin down a particular distribution. Once n and p are known, every binomial probability, the mean and the standard deviation can be computed.
Category: Binomial Distribution
Question 10
A random variable whose possible values can be listed and counted, such as the number of absences in a week, is called discrete.
Missing Word
discrete
Discrete random variables take separate values, usually whole-number counts. Their distributions assign a probability to each possible value.
Category: Probability Distributions
Question 11
A random variable that can take any value in an interval, such as the time needed to finish a race, is called continuous.
Missing Word
continuous
Continuous random variables come from measurements like time, height or weight, where any value in a range is possible. They are handled with density curves rather than a list of separate probabilities.
Category: Probability Distributions
Question 12
For X ~ B(10, 0.1), most of the probability sits near 0 and a long tail stretches toward larger values, so the distribution is skewed to the right.
Missing Word
skewed
A distribution is skewed when one tail is longer than the other. With p = 0.1 the mean np is only 1, values near 0 are most likely, and values up to 10 form a long right tail. A binomial distribution is symmetric only when p = 0.5.
Category: Binomial Distribution
Question 13
The expected value of a random variable, such as the roll of a die, is a weighted average of its values, using their probabilities as the weights.
Missing Word
weighted
Each value is multiplied by its probability before adding, so likely values count more than unlikely ones. That weighting is what makes the expected value differ from a plain average of the possible values.
Category: Mean and Standard Deviation
Question 14
A game whose expected value for the player is exactly 0 is called fair, since neither the player nor the house gains on average.
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fair
In a fair game the long-run average gain is zero. A negative expected value favors the house, and a positive one favors the player.
Category: Mean and Standard Deviation
Question 15
Drawing items from a group and putting each one back before the next draw is called sampling with replacement.
Missing Word
replacement
Sampling with replacement restores the group after every draw, so each draw has the same probabilities and the draws are independent, as a binomial setting requires. Sampling without replacement changes the group after each draw, which makes the draws dependent and leads to the hypergeometric distribution.
Category: Hypergeometric Distribution