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Chapter 7 Discrete Distributions

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Random Variable - A numerical variable whose value depends on the outcome of a chance experiment

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Two types: Discrete – count of some random variable Continuous – measure of some random variable

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A random variable is discrete if its set of possible values is a collection of isolated points on the number line. Discrete and Continuous Random Variables A random variable is continuous if its set of possible values includes an entire interval on the number line. We will use lowercase letters, such as x and y, to represent random variables. Possible values of a discrete random variable Possible values of a continuous random variable

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Examples 1.Experiment: A fair die is rolled Random Variable: The number on the up face Type: Discrete 2.Experiment: A coin is tossed until the 1 st head turns up Random Variable: The number of the toss that the 1 st head turns up Type: Discrete

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3.Experiment: Choose and inspect a number of parts Random Variable: The number of defective parts Type: Discrete 4.Experiment: Measure the voltage in a outlet in your room Random Variable: The voltage Type: Continuous 5.Experiment: Observe the amount of time it takes a bank teller to serve a customer Random Variable: The time Type: Continuous

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Discrete Probability Distribution 1)Gives the probabilities associated with each possible x value 2)Usually displayed in a table, but can be displayed with a histogram or formula

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Discrete probability distributions 3)For every possible x value, 0 < P(x) < 1. 4) For all values of x, P(x) = 1.

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Suppose you toss 3 coins & record the number of heads. What is the sample space?

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The Random Variable X is defined as … The number of heads tossed # of heads X=0 TTT X=1 HTT,THT,TTH X=2 HHT,HTH,THH X=3 HHH

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Create a probability distribution. Create a probability histogram. X0123X0123 P(X)

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Create a probability distribution. X0123X0123 P(X) Now we can use the probability distribution table to answer questions about the variable X. What is the probability of getting exactly 2 heads? What is the probability of getting at least 2 heads? What is the probability of getting at least one head? P(X=2) =.375 P(X>2) = P(X=2) + P(X=3) =.5 P(X>1) = P(X=1) + P(X=2) + P(X=3) =.875

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Let x be the number of courses for which a randomly selected student at a certain university is registered. X P(X) ? P(x = 4) = P(x < 4) = What is the probability that the student is registered for at least five courses? Why does this not start at zero? P(x > 5) =.61

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Mean and Variance of Discrete Random Variables

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Probability Distributions are also described by measures of central tendency and variability. The MEAN of a discrete random variable X is the average of the possible outcomes of X WITH the weights (probabilities). Other names for the MEAN are the WEIGHTED AVERAGE or the EXPECTED VALUE.

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Probability Distributions are also described by measures of central tendency and variability. The VARIANCE is an average of the squared deviation of the values of the variable X from its mean. The STANDARD DEVIATION is the square root of the variance.

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Formulas for mean & variance Found on formula card!

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Let x be the number of courses for which a randomly selected student at a certain university is registered. X P(X) What is the mean and standard deviations of this distribution? 4.66 & =

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Find the mean and standard deviation for the number of heads out of 3 tosses. X0123X0123 P(X) & =.866

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Heres a game: If a player rolls two dice and gets a sum of 2 or 12, he wins $20. If he gets a 7, he wins $5. The cost to roll the dice one time is $3. Is this game fair? A fair game is one where the cost to play EQUALS the expected value!

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If a player rolls two dice and gets a sum of 2 or 12, he wins $20. If he gets a 7, he wins $5. The cost to roll the dice one time is $3. Is this game fair? X0520 P(X)7/91/61/18 NO, since = $1.944 which is less than it cost to play ($3). What is the random variable X? Make a probability distribution table.

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Lets play a game. A player pays $5 and draws a card from a deck. If he draws the ace of hearts, he is paid $100. For any other ace, he is paid $10, and for any other heart, he is paid $5. If he draws anything else, he gets nothing. Would you be willing to play? What is the random variable X? _____________________ What are the values for the random variable? ____________ OutcomePayout (X)ProbabilityE(X)Deviation

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An insurance company offers a death and disability policy that pays $10,000 when you die or $5000 if you are permanently disabled. It charges a premium of on $50 a year for this benefit. Is the company likely to make a profit selling such a plan?

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Policyholder Outcome Payout X Probability P(x) Death$10,000 Disability$5000 Neither$0 Suppose that the death rate in any year is 1 out of every 1000 people, and that another 2 out of 1000 suffer some kind of disability.

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Policyholder Outcome Payout X Probability P(x) Death$10,000 Disability$5000 Neither$0 E (X) = Expected value = μ = = $10,000( ) + $5000( ) + $0( ) = $10 + $10 + $0 = $20

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Policyholder Outcome Payout X Probability P(x) Death$10,000 Disability$5000 Neither$0 Deviation (x – μ) (10,000 – 20) = 9980 (5000 – 20) = 4980 (0 – 20) = - 20 Var(x) = ( ) ( ) + (-20) 2 ( ) = 149,600 Standard deviation (X) = 149,600 = $386.78

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Linear function of a random variable If x is a random variable and a and b are numerical constants, then the random variable y is defined by and The mean is changed by addition & multiplication! ONLY The standard deviation is ONLY changed by multiplication!

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Let x be the number of gallons required to fill a propane tank. Suppose that the mean and standard deviation is 318 gal. and 42 gal., respectively. The company is considering the pricing model of a service charge of $50 plus $1.80 per gallon. Let y be the random variable of the amount billed. What is the mean and standard deviation for the amount billed? = $ & = $75.60

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Linear combinations Just add or subtract the means! add If independent, always add the variances!

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The mean of the sum of two random variables is the sum of the means. E(X + Y) = E(X) + E(Y) The mean of the difference of two random variables is the difference of the means. E(X - Y) = E(X) - E(Y) If the random variable are independent, the variance of their sums OR difference is always the sum of the variances. Var(X + Y) = Var(X) + Var(Y)

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MeanSD X102 Y205 What is the mean and standard deviation of: a) 3XMean = 30; standard deviation = 6 b) Y + 6Mean = 26; standard deviation = 5 c) X + Y d) X - Y e) X 1 + X 1

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A nationwide standardized exam consists of a multiple choice section and a free response section. For each section, the mean and standard deviation are reported to be meanSD MC386 FR307 If the test score is computed by adding the multiple choice and free response, then what is the mean and standard deviation of the test? 68 & =

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Example Suppose x is the number of sales staff needed on a given day. If the cost of doing business on a day involves fixed costs of $255 and the cost per sales person per day is $110, find the mean cost (the mean of x or x ) of doing business on a given day where the distribution of x is given below.

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Example continued We need to find the mean of y = x

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Example continued We need to find the variance and standard deviation of y = x

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