Review of Discrete Probability Distributions
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| What is a “random variable”? | A random variable is a numerical measure of the outcome from a probability experiment, so its value is determined by chance. Random variables are denoted using letters such as X.
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| What is a “discrete random variable”? | A discrete random variable has either a finite or countable number of values. The values of a discrete random variable can be plotted on a number line with space between each point.
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| What is a “continuous random variable”? | A continuous random variable has infinitely many values. The values of a continuous random variable can be plotted on a line in an uninterrupted fashion.
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| How is “Probability Distribution” defined? | A probability distribution provides the possible values of the random variable X and their corresponding probabilities. A probability distribution can be in the form of a table, graph or mathematical formula.
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| State the required rules for a “Discrete Probability Distribution”? | If P(x) denotes the probability that the discrete random variable X equals a particular value “x”, then for the model to be a Discrete Probability Distribution: 1)Σ P(x) =1 and 2) 0 ≤ P(x) ≤ 1.
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| Define a “discrete probability histogram”. | A discrete probability histogram is a histogram in which the horizontal axis corresponds to the value of the random variable and the vertical axis represents the probability of that value of the random variable.
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| Give the formula for the “mean” of a discrete random variable. | The mean of a discrete random variable is given as: µx = Σ [x∙ P(x)] where x is the value of the random variable and P(x) is the probability of observing that value of x.
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| Give the formula for the “variance” of a discrete random variable. | The formula for the “variance” of a discrete random variable is given as: σ^2 = Σ[(x - µ)^2 ∙ P(x)] where x is the value of the random variable and P(x) is the probability of observing that value of x.
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| How would determine the formula for the standard deviation of a discrete random variable? | We would determine the formula for the standard deviation of a discrete random variable by taking the square root of the variance.
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| Define a “Binomial Probability Experiment”. | An experiment performed a fixed number of times (“n”), called a trials. The trials are independent with two mutually exclusive (or disjoint) outcomes -- success or failure. The probability of success is fixed for each trial of the experiment.
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| What type of probability distribution is a “Binomial Probability Distribution” – discrete or continuous? Why? | The Binomial distribution is a discrete probability distribution. This is because with the binomial distribution you are “counting” the number of successes, “x”, in “n” trials.
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| Why is this type of distribution called a binomial distribution? | This type of distribution is called a binomial distribution because for each trial, there are two mutually exclusive (or disjoint) outcomes -- success or failure.
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| What condition is required concerning the trials of a binomial experiment? | The trials of a binomial experiment must be independent. This means the outcome of one trial will not affect the outcome of the other trials.
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| Is it acceptable for the probability of success to be different from trial to trial in a binomial experiment? | No. The probability of success for each trial must be the same to qualify as a binomial experiment
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| Specify the notation used in a Binomial Probability Distribution. | The letter “n” denotes the number of trials of the experiment; “p” denotes the probability of success (1 – p is the probability of failure); “X” denotes the number of successes in “n” trials of the experiment. So, 0 < x < n.
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| Give the formula for the “mean” of a Binomial random variable. | The “mean” of a Binomial random variable is given by the formula: µx = np; where “n” is the number of trials and “p” is the probability of success for each trial.
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| Give the formula for the “standard deviation” of a Binomial random variable. | The “standard deviation” of a Binomial random variable is given by the formula: σx = sqrt(np(1 – p)
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| Suppose X is a binomial random variable and the number or trials, n = 18 and the probability of success on each trial is 0.25. What is the probability of exactly x = 10 successes? | We are looking for P(X = 10). Using the TI-83/84 function “binompdf (n, p, x)” (Notice the PDF)we have: binompdf(18, 0.25, 10) = 0.0042
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| Suppose X is a binomial random variable and the number or trials, n = 18 and the probability of success on each trial is 0.25. What is the probability of x = 5 or less successes? | We are looking for P(X ≤ 5). Using the TI-83/84 function “binomcdf (n, p, x)” (Notice the CDF)we have: binomcdf(18, 0.25, 5) = 0.7175
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| Suppose X is a binomial random variable and the number or trials, n = 18 and the probability of success on each trial is 0.25. What is the probability of more than x = 5 successes? | We are looking for P(X > 5). Which is the same thing as 1 – P(X ≤ 5 ). Using the TI-83/84 function “binomcdf (n, p, x)” (Notice the CDF). Thus, 1 – binomcdf(18, 0.25, 5) = 1 – 0.7175 = 0.2825
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| In general, what would we expect the shape of a binomial distribution to be if the probability of success for each trial, p < 0.5? | In general, if the probability of success, p < 0.5, we would expect the shape of the binomial distribution to be skewed right.
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| In general, what would we expect the shape of a binomial distribution to be if the probability of success for each trial, p > 0.5? | In general, if the probability of success, p > 0.5, we would expect the shape of the binomial distribution to be skewed left.
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| What would we expect the shape of a binomial distribution to be if the probability of success for each trial, p = 0.5? | If the probability of success, p = 0.5, we would expect the shape of the binomial distribution to be symmetric (approximately bell-shaped).
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| Regardless of the probability of success for each trial, p, what can we assume about the shape of the binomial distribution if np(1 – p) ≥ 10? | Approximately bell-shaped, because, as “n”, increases, the probability distribution of the binomial random variable X becomes bell shaped. So, if “n” is large enough for np(1 – p) ≥ 10, the binomial distribution is approximately bell-shaped.
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