3 Discrete Random Variables and Probability Distributions Copyright
3 Discrete Random Variables and Probability Distributions Copyright Cengage Learning. All rights reserved. 1 3.2
Probability Distributions for Discrete Random Variables Copyright Cengage Learning. All rights reserved. 2 Probability Distributions for Discrete Random Variables Probabilities assigned to various outcomes in in turn determine probabilities associated with the values of any particular rv X.
The probability distribution of X says how the total probability of 1 is distributed among (allocated to) the various possible X values. Suppose, for example, that a business has just purchased four laser printers, and let X be the number among these that require service during the warranty period. 3 Probability Distributions for Discrete Random Variables Possible X values are then 0, 1, 2, 3, and 4. The probability distribution will tell us how the probability of 1 is subdivided
among these five possible values how much probability is associated with the X value 0, how much is apportioned to the X value 1, and so on. We will use the following notation for the probabilities in the distribution: p (0) = the probability of the X value 0 = P(X = 0) p (1) = the probability of the X value 1 = P(X = 1) and so on. In general, p (x) will denote the probability assigned to the value x. 4
Example 3.7 The Cal Poly Department of Statistics has a lab with six computers reserved for statistics majors. Let X denote the number of these computers that are in use at a particular time of day. Suppose that the probability distribution of X is as given in the following table; the first row of the table lists the possible X values and the second row gives the probability of each such value.
5 Example 3.7 contd We can now use elementary probability properties to calculate other probabilities of interest. For example, the probability that at most 2 computers are in use is P(X 2) = P(X = 0 or 1 or 2) = p(0) + p(1) + p(2)
= .05 + .10 + .15 = .30 6 Example 3.7 contd Since the event at least 3 computers are in use is complementary to at most 2 computers are in use,
P(X 3) = 1 P(X 2) = 1 .30 = .70 which can, of course, also be obtained by adding together probabilities for the values, 3, 4, 5, and 6. 7 Example 3.7 contd
The probability that between 2 and 5 computers inclusive are in use is P(2 X 5) = P(X = 2, 3, 4, or 5) = .15 + .25 + .20 + .15 = .75 whereas the probability that the number of computers in use is strictly between 2 and 5 is P(2 < X < 5) = P(X = 3 or 4) = .25 + .20 = .45
8 Probability Distributions for Discrete Random Variables Definition 9 Probability Distributions for Discrete Random Variables In words, for every possible value x of the random variable, the pmf specifies the probability of observing that value
when the experiment is performed. The conditions p (x) 0 and all possible x p (x) = 1 are required of any pmf. The pmf of X in the previous example was simply given in the problem description. We now consider several examples in which various probability properties are exploited to obtain the desired distribution. 10 A Parameter of a Probability
Distribution 11 A Parameter of a Probability Distribution The pmf of the Bernoulli rv X in Example 3.9was p(0) = .8 and p(1) = .2 because 20% of all purchasers selected a desktop computer. At another store, it may be the case that p(0) = .9 and p(1) = .1. More generally, the pmf of any Bernoulli rv can be
expressed in the form p (1) = and p (0) = 1 , where 0 < < 1. Because the pmf depends on the particular value of we often write p (x; ) rather than just p (x): (3.1) 12 A Parameter of a Probability Distribution Then each choice of a in Expression (3.1) yields a different pmf. Definition
13 A Parameter of a Probability Distribution The quantity in Expression (3.1) is a parameter. Each different number between 0 and 1 determines a different member of the Bernoulli family of distributions. 14 Example 3.12 Starting at a fixed time, we observe the gender of each newborn child at a certain hospital until a boy (B) is born.
Let p = P (B), assume that successive births are independent, and define the rv X by x = number of births observed. Then p(1) = P(X = 1) = P(B) =p 15 Example 3.12
16 Example 3.12 contd Continuing in this way, a general formula emerges: (3.2) The parameter p can assume any value between 0 and 1.
Expression (3.2) describes the family of geometric distributions. In the gender example, p = .51 might be appropriate, but if we were looking for the first child with Rh-positive blood, then we might have p = .85. 17 The Cumulative Distribution Function 18
The Cumulative Distribution Function For some fixed value x, we often wish to compute the probability that the observed value of X will be at most x. For example, let X be the number of number of beds occupied in a hospitals emergency room at a certain time of day; suppose the pmf of X is given by Then the probability that at most two beds are occupied is 19
The Cumulative Distribution Function Furthermore, since X 2.7 if and only if X 2, we also have P(X 2.7) = .75, and similarly P(X 2.999) = .75. Since 0 is the smallest possible X value, P(X -1.5) = 0, P(X -10) = 0, and in fact for any negative number x, P(X x) = 0. And because 4 is the largest possible value of X, P(X 4) = 1, P(X 9.8) = 1, and so on. 20
The Cumulative Distribution Function Very importantly, because the latter probability includes the probability mass at the x value 2 whereas the former probability does not. More generally, P(X x) P(X x) whenever x is a possible value of X. Furthermore, P(X x) is a well-defined and computable probability for any number x. 21
The Cumulative Distribution Function Definition 22 Example 3.13 A store carries flash drives with either 1 GB, 2 GB, 4 GB, 8 GB, or 16 GB of memory. The accompanying table gives the distribution of Y = the amount of memory in a purchased drive:
23 Example 3.13 contd Lets first determine F (y) for each of the five possible values of Y: F (1) = P (Y 1) = P (Y = 1)
= p (1) = .05 F (2) = P (Y 2) = P (Y = 1 or 2) = p (1) + p (2) = .15 24 Example 3.13 contd
Example 3.13 contd Now for any other number y, F (y) will equal the value of F at the closest possible value of Y to the left of y. For example, F(2.7) = P(Y 2.7) = P(Y 2) = F(2)
= .15 F(7.999) = P(Y 7.999) = P(Y 4) = F(4) = .50 26 Example 3.13 contd If y is less than 1, F (y) = 0 [e.g. F(.58) = 0], and if y is at
least 16, F (y) = 1[e.g. F(25) = 1]. The cdf is thus 27 Example 3.13 contd A graph of this cdf is shown in Figure 3.5. A graph of the cdf of Example 3.13
Figure 3.13 28 The Cumulative Distribution Function For X a discrete rv, the graph of F (x) will have a jump at every possible value of X and will be flat between possible values. Such a graph is called a step function. Proposition 29
The Cumulative Distribution Function The reason for subtracting F (a)rather than F (a) is that we want to include P(X = a) F (b) F (a); gives P (a < X b). This proposition will be used extensively when computing binomial and Poisson probabilities in Sections 3.4 and 3.6. 30 Example 3.15 Let X = the number of days of sick leave taken by a
randomly selected employee of a large company during a particular year. If the maximum number of allowable sick days per year is 14, possible values of X are 0, 1, . . . , 14. 31 Example 15 contd
Encourage practitioners to review chart history to detect possible fraudulent activity. Review SB 459 requirements for prescribers. Provide registration instructions for prescribers not registered with the PMP. Letters to Occupational Licensing Boards.
Measurement, Modeling and Analysis of a Peer-to-Peer File-Sharing Workload Krishna Gummadi, Richard Dunn, Stefan Saroiu Steve Gribble, Hank Levy, John Zahorjan Most of these are taken from the original powerpoint presentation by Gummadi
The "New Hume" is the view of John Wright, Galen Strawson, Peter Kail and others that Hume is instead a "causal realist". The Idea of Cause In Treatise I iii 2, Hume identifies the components of the idea of causation...
Rationale: Candidates will understand and apply the competencies reflected in the NAEYC standards when they are able to observe, implement, and receive constructive feedback in real-life settings. Criterion 5: Quality of Field Experiences
simple instructions all 32 bits wide very structured, no unnecessary baggage only three instruction formats op rs rt rd shamt funct op rs rt 16 bit address op 26 bit address R I J Review of MIPS Instruction Formats MIPS...
Use the dropdown menus when necessary. Enter alternative date ranges when you would be available for this course Return to Table of Contents Step 16 How to Apply for Training Select more classes here At the bottom of the application...
Ready to download the document? Go ahead and hit continue!