Chap5

A RANDOM VARIABLE is a quantitative variable whose value depends on chance. A discrete random variable is a random variable whose possible values form a FINITE (or COUNTABLY INFINITE) set of numbers. What does the probability distribution of a discrete random variable tell us? THE POSSIBLE VALUES AND CORRESPONDING PROABILITIES OF THE DISCRETE RANDOM VARIABLE. How do we graphically portray the probability distribution of a discrete random variable? PROBABILITY HISTOGRAM. If we sum the probabilities of the possible values of a discrete random variable, the result is always equal to 1. A random variable X takes the value 2 with probability 0.386. Express that fact using probability notation. P(X=2) = 0.386. If we make repeated independent observations of the random variable X, then in approximately what percentage of those observations will we observe the value 2. 38.6% Roughly how many times would we expect to observe the value 2 in 50 observation? 19.3% 500 observations? 193. A random variable X has mean 3.6. If we make a large number of repeated independent observations of the random variable X, then the average value of those observations will be approximately equal to 3.6. Two random variables, X and Y, have standard deviations 2.4 and 3.6, respectively. Which one is more likely to take a value close to it mean? X, Explain your answer. Because it has a smaller standard deviation, therefore less variation. List the three requirements for repeated trials to constitute Bernoulli trials. Each trial has the same two possible outcomes; the trials are independent; the probability of a success remains the same from trial to trial. What is the relationship between Bernoulli trials and the binomial distribution? The binomial distribution is the probability distribution for the number of successes in a sequence of Bernoulli trials. In 10 Bernoulli trials, how many outcomes contain exactly three successes? 10 nCr 3 120. Explain how the special formulas for the man and standard deviation of a binomial or Poisson random variable are derived. Substitute the binomial (or Poisson) probability formula into the formulas for the mean and standard deviation of a discrete random variable and then simplify mathematically. Suppose a simple random sample of size n is taken from a finite population in which the proportion of members having a specified attribute is p. Let X be the number of members sampled that have the specified attribute. If the sampling is dome with replacement, identify the probability distribution of X. a binomial distribution. If the sampling is dome without replacement, identify the probability distribution of X. hypergeometric distribution. Under what conditions is it acceptable to approximate the probability distribution is part (b) by the probability distribution in part (a)? When the ample size does not exceed 5% of the population size. Why is it acceptable? Because, under this condition, there is little difference between sampling with and without replacement.

Class Level	No. of students
1	6,382
2	6,312

8,157

4	11,686
	32,438 total

Probability distribution of the Probability histogram for the random variable X.

random variable X.

X	P(X = x)
1	6383/32438= .196
2	6,312/32,428= .194
3	8,157/32,438= .251
4	11,686/32,438=.359

The game of craps is played by rolling two balanced dice. A first roll of a sum of 7 or 11wins; and a first roll of a sum of 2, 3, or 12 loses. To win with any other first sum, that sum must be repeated before a sum of 7 is thrown. If can be shown that the probability is 0.493 that a player wins a game of craps. Suppose we consider a win by a player to be a success, s.

What is the probability, p? p = 0.493. Construct a table showing the possible win-lose results and their probabilities for three games of craps. Round each probability to three decimal places. List the outcomes in which the player wins exactly two out of three times. Ssf, sfs, fss Determine the probability of each of the outcomes in part d. 0.123. Each probability is obtained by multiplying two success probabilites of 0.493 and one failure probability of 0.507. Find the probability that the player wins exactly two out of three times. 0.369.