Three performers are invited to an audition. Assume all three performers arrive on their own. Here, the probability of any one of them being late is 0.2. Let X be the number of late participants of the seminar. (a) How do I find the distribution (pmf) of X? (b) How would I calculate the mean and variance of X?
Three performers are invited to an audition. Assume all three performers arrive on their own. Here, the probability of any one of them being late is 0.2. Let X be the number of late participants of the seminar. (a) How do I find the distribution (pmf) of X? (b) How would I calculate the mean and variance of X?
Three performers are invited to an audition. Assume all three performers arrive on their own. Here, the probability of any one of them being late is 0.2. Let X be the number of late participants of the seminar. (a) How do I find the distribution (pmf) of X? (b) How would I calculate the mean and variance of X?
Three performers are invited to an audition. Assume all three performers arrive on their own. Here, the probability of any one of them being late is 0.2. Let X be the number of late participants of the seminar.
(a) How do I find the distribution (pmf) of X?
(b) How would I calculate the mean and variance of X?
Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...
Expert Solution
Step 1: Basic Idea:
Identify the random variable: In this problem, the random variable is X, which represents the number of late participants among the three performers.
Determine the probability of success and failure: You are given that the probability of any one performer being late is 0.2, so p (probability of success) is 0.2, and (1 - p) is the probability of not being late, which is 0.8.
Recognize the binomial distribution: Since each performer's lateness is independent, and you want to find the probability distribution of the number of late performers out of a fixed number of trials (in this case, 3 trials), you can use the binomial distribution.
Calculate the probability mass function (pmf): Use the binomial distribution formula to calculate the probabilities of different values of X (the number of late participants) for k = 0, 1, 2, and 3.
Since the distribution is binomial.
Formulae for mean and variance of X:
E(X) = np ; n -> number of trials ; k = 0,1,...,n ; here n = 3
