Please explain! Give thorough explanation! Give a good examples of : Permutation
ID: 3174975 • Letter: P
Question
Please explain! Give thorough explanation! Give a good examples of: Permutations and combinations; Conditional probability; Independent events; Random variables; Probability distributions and densities; Expectation; Moments; Moment generating functions; Functions of random variables; Central Limit Theorem; Sampling; Confidence intervals; The gamma, chi-square, T, F, and bivariate normal distributions; Central Limit Theorem; confidence intervals and tests of hypothesis; the Neymen-Pearson Theorem; likelihood ratio test; estimation; sufficiency, unbiasedness, completeness; the Rao- Blackwell Theorem; the Rao-Cramer inequality; the method of maximum likelihood; the chi-square test; introduction to the analysis of variance and regression. Thank you very much in advance! Much karma to you my friend ))) PLEASE PLEASE PLEASE NOTE: Thorough explanation and examples are the keys...
Explanation / Answer
1) A permutation is an arrangement of all or part of a set of objects, with regard to the order of the arrangement. nPr = n(n - 1)(n - 2) ... (n - r + 1) = n! / (n - r)!
For example, suppose we have a set of three letters: A, B, and C. We might ask how many ways we can arrange 2 letters from that set. Each possible arrangement would be an example of a permutation. The complete list of possible permutations would be: AB, AC, BA, BC, CA, and CB.
2) A combination is a selection of all or part of a set of objects, without regard to the order in which objects are selected. nCr = n(n - 1)(n - 2) ... (n - r + 1)/r! = n! / r!(n - r)! = nPr / r!
For example, suppose we have a set of three letters: A, B, and C. We might ask how many ways we can select 2 letters from that set. Each possible selection would be an example of a combination. The complete list of possible selections would be: AB, AC, and BC.
3)The conditional probability of an event B is the probability that the event will occur given the knowledge that an event A has already occurred. This probability is written P(B|A), notation for the probability of B given A. In the case where events A and B are independent (where event A has no effect on the probability of event B), the conditional probability of event B given event A is simply the probability of event B, that is P(B)
Example 1. In a group of 100 sports car buyers, 40 bought alarm systems, 30 purchased bucket seats, and 20 purchased an alarm system and bucket seats. If a car buyer chosen at random bought an alarm system, what is the probability they also bought bucket seats?
Step 1: Figure out P(A). It’s given in the question as 40%, or 0.4.
Step 2: Figure out P(AB). This is the intersection of A and B: both happening together. It’s given in the question 20 out of 100 buyers, or 0.2.
Step 3: Insert your answers into the formula:
P(B|A) = P(AB) / P(A) = 0.2 / 0.4 = 0.5.
The probability that a buyer bought bucket seats, given that they purchased an alarm system, is 50%
4)independent events - When two events are said to be independent of each other, what this means is that the probability that oneevent occurs in no way affects the probability of the other event occurring. An example of two independent events is as follows;
For example, if I roll a standard six-sided die and flip a coin, the two events will not have any effect on the probability of the other. Regardless of the outcome of rolling the die, the coin will be just as likely to land on heads or tails. Likewise, regardless of the outcome of the coin flip, the die will be just as likely to land on one of the six numbers of the die.
5) A random variable is a variable whose value is unknown or a function that assigns values to each of an experiment's outcomes. Random variables are often designated by letters and can be classified as discrete, which are variables that have specific values, or continuous, which are variables that can have any values within a continuous range.
example. We'll start with tossing coins. I want to know how many heads I might get if I toss two coins. Since I only toss two coins, the number of heads I could get is zero, one, or two heads. So, I define X (my random variable) to be the number of heads that I could get. For example: number of marbles in a jar, number of students present or number of heads when tossing two coins.
6)probability density function - A continuous random variable takes on an uncountably infinite number of possible values. For a discrete random variable X that takes on a finite or countably infinite number of possible values, we determined P(X = x) for all of the possible values of X, and called it the probability mass function ("p.m.f."). For continuous random variables, as we shall soon see, the probability that X takes on any particular value x is 0. That is, finding P(X = x) for a continuous random variable X is not going to work. Instead, we'll need to find the probability that X falls in some interval (a, b), that is, we'll need to find P(a < X < b). We'll do that using a probability density function ("p.d.f."). We'll first motivate a p.d.f. with an example, and then we'll formally define it.
Example - Even though a fast-food chain might advertise a hamburger as weighing a quarter-pound, you can well imagine that it is not exactly 0.25 pounds. One randomly selected hamburger might weigh 0.23 pounds while another might weigh 0.27 pounds. What is the probability that a randomly selected hamburger weighs between 0.20 and 0.30 pounds we have to calculate.
7) moments - you’ll probably come across something that states the first moment is the mean or that the second measures how wide a distribution is (the variance). Loosely, these definitions are right. Technically, a moment is defined by a mathematical formula that just so happens to equal formulas for some measures in statistics. formula for moments The sth moment = (x1s + x2s + x3s + . . . + xns)/n.
8) The moment generating function (MGF) of a random variable XX is a function MX(s)MX(s) defined as
MX(s)=E[esX].MX(s)=E[esX].
We say that MGF of XX exists, if there exists a positive constant aa such that MX(s)MX(s) is finite for all s[a,a]s[a,a].
9) A confidence interval is a range of values that describes the uncertainty surrounding an estimate. We indicate a confidence interval by its endpoints; for example, the 90% confidence interval for the number of people, of all ages, in poverty in the United States in 1995 (based on the March 1996 Current Population Survey) is "35,534,124 to 37,315,094." A confidence interval is also itself an estimate. It is made using a model of how sampling, interviewing, measuring, and modeling contribute to uncertainty about the relation between the true value of the quantity we are estimating and our estimate of that value.