Zarestky Math 141 Week in Review Texas A&M University 1 Week 9 Key Topics 8.4 The Binomial Distribution Bernoulli or Binomial Experiment Properties • The number of trials in the experiment is fixed. • There are two outcomes of the experiment: success and failure. • The probability of success in each trial is the same. • The trials are independent of each other. Computing Binomial Probabilities • In a binomial experiment, in which the probability of success in any trial is p and the probability of failure in any trial is q = 1 – p, the probability of exactly r successes in n independent trials is P X = r( )= C n ,r( )prqn! r. • X is called the binomial random variable and the probability distribution of X is the binomial distribution. For a binomial probability question you must do the following: • Decide that it is a Bernoulli trial by checking the 4 criteria. • Define success. • Find the number of times the experiment is performed, n. • Find the probability of success, p. • Determine the desired number of successes, r. If X is a binomial random variable associated with a binomial experiment consisting of n trials with probability of success p and probability of failure q = 1 − p, then • mean µ= E X( )= np • variance Var X( )= npq • standard deviation !X= npq 8.5 The Normal Distribution Continuous Probability Distribution • Recall that a continuous random variable can take on any value. • A continuous probability distribution is a probability distribution associated with a continuous random variable. o Defined by a function f whose domain is the interval of real values taken on by the random variable X associated with the experiment. Probability Density Function (pdf) • f(x), the continuous probability distribution, is nonnegative for all values of x • The area of the region between the graph of f and the x-axis is equal to 1. • P(a < X < b) is the probability that the random variable X assumes a value between a and b, (a < x < b). o Given by the area of the region between f(x) and the x-axis from x = a to x = b.Zarestky Math 141 Week in Review Texas A&M University 2 o If you have had calculus before, this is f x( )ab!dx.Zarestky Math 141 Week in Review Texas A&M University 3 Graphs of Probability Distributions • Discrete finite variables: Graph the probability as a histogram. o Each rectangle has a base of width 1 (centered on the values of the r. v.) and a height equal to the probability of that value of the r. v. o The area, length × height, is the probability that a given value of the r. v. occurs. o To find the probability of a range of X values, add up the areas over the range of X values. • Continuous variables: Graph the probability distribution as a smooth curve. o The area above the range of X values is the probability of these X values occurring. o Many natural sciences, behavioral sciences, psychological test scores, and physical phenomena have a continuous distribution with a bell-shaped curve. o These special families of important continuous probability distributions are known as the normal distributions. o The standard normal distribution has mean µ = 0 and σ = 1. • Use Z instead of X as the continuous random variable. Properties of Normal Curves • The curve has a peak at x = µ and is symmetric with respect to the vertical line x = µ. • The curve always lies above the x-axis but approaches the x-axis as x extends infinitely in either direction. • The area under the entire curve is 1. • 68.27% of the area under the curve lies within 1 standard deviation of the mean (that is, between µ – σ and µ + σ). • 95.45% of the area lies within 2 standard deviations of the mean. • 99.73% of the area lies within 3 standard deviations of the mean. • The shape is completely determined by µ and σ. 8.6 Applications of the Normal Distribution The Normal Curve Approximation to the Binomial Distribution • We will use the normal curve to approximate the binomial distribution. • As the number of trials increases, the binomial distribution comes closer and closer to the normal curve approximation. • If you are asked to approximate a given binomial probability with n trials, probability of success p, and probability of failure q, you will use a normal distribution with µ = np and σ = npq. • To determine the boundaries for the normal curve approximation, identify the rectangles in the binomial histogram that should be included. Then adjust the boundaries by ±0.5 to move to the left or right endpoint of a
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