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Ch6_Discete PDFs

Review of Discrete Probability Distributions

QuestionAnswer
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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
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.
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.
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)
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
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
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
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.
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.
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).
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.
Created by: wgriffin410