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Stat 330 (Spring 2015): slide set 7 1 for the event {ω|ω ∈ Ω and X(ω) = x}. {X = x} Notation: To avoid cumbersome notation, we write That is, the set of possible values for X(ω) is {0, 1, 2, 3} 2 X(ω) is then an integer between 0 and 3 for every possible sequence of throws of 3 darts. X(ω) = k, if the outcome ω has k hits to the red area, and 3 − k hits to the gray area. More formally: • Between 2 and 7 wrong bits (inclusive) received: {2 ≤ X ≤ 7} • Exactly 2 wrong bits received: {X = 2} • At least one wrong bit received: {X ≥ 1} • No wrong bits received: {X = 0} Examples of events and equivalent expressions using X. That is, the possible values for X are {0, 1, 2, 3, 4, 5, 6, 7, 8} 3 Use random variable X to “count” the number of wrong bits received. X assigns a value between 0 and 8 to each sequence in the sample space. Example Suppose, 8 bits are sent through a communication channel. Each bit has a certain probability to be received incorrectly. We are interested in the number of bits that are received incorrectly. on any throw. We deﬁne X to be the function that assigns the number of times that the red area is hit in a sequence of three throws. 1 9 Practice with notation Stat 330 (Spring 2015): slide set 7 Also note that P (red) = Imagine we throw three darts on this board one by one and we are interested in the number of times the red area has been hit. This count is a random variable! Example: Very simple Dartboard Random Variable A function X : Ω → R is called a random variable. Dartboard (continued...) Last update: January 16, 2015 Stat 330 (Spring 2015) Slide set 7 Stat 330 (Spring 2015): slide set 7 Intuitive idea: If the value of a numerical variable depends on the outcome of an experiment, we call the variable a random variable. Random Variables Stat 330 (Spring 2015): slide set 7 is deﬁned as (ii) i pX (xi ) = 1 (the sum of all values is 1) 6 (i) 0 ≤ pX (x) ≤ 1 for all x ∈ {x1, x2, x3, . . .} (all values must be between 0 and 1) pX is the pmf of X, if and only if Properties of PMF: 3. 2. 1. z pZ (z) y pY (y) x pX (x) 0 0.22 -1 0.1 -3 0.1 1 0.18 0 0.45 -1 0.45 3 0.24 1.5 0.25 0 0.15 5 0.17 3 -0.05 5 0.25 7 0.18 4.5 0.25 7 0.05 7 Experiment: Which of the following functions is a valid probability mass function? These properties give us an easy method to check, whether a function is a probability mass function Examples: · 19 · 89 + 19 · 89 · 19 + 89 · 19 · 19 =? Stat 330 (Spring 2015): slide set 7 1 9 Stat 330 (Spring 2015): slide set 7 = = P (RRG) + P (RGR) + P (GRR) = P (RRG or RGR or GRR) P (2 reds) = P (X = 2) The event “exactly 2 reds” is formally written as {ω : X(ω) = 2}, and with the simpler notation for this event is X = 2. Example: Dartboard What is the probability that you hit the red square exactly twice in 3 throws? • Very often we are interested in probabilities of the form P (X = x). We can think of this expression as a function, that yields diﬀerent probabilities depending on the value of x. 5 The function pX (x) := P (X = x) is called the probability mass function of X. Deﬁnition: Probability Mass Function, PMF Probability Mass Function Stat 330 (Spring 2015): slide set 7 • Assume X is a discrete random variable. The image of X is therefore countable and can be written as {x1, x2, x3, . . .} Discrete R.V.s 4 Im(X) = {0, 1, 2, 3, 4, 5, 6, 7, 8} is a ﬁnite set → X is a discrete random variable. 2. Communication channel: X = “# of incorrectly received bits” image of Y is an interval (uncountable image) → Y is a continuous random variable. Im(Y ) = (0, ∞). 1. Put a disk drive into service, measure Y = “time till the ﬁrst major failure”. Deﬁnition: The image of a random variable X Im(X) := {x : x = X(ω) for some ω ∈ Ω}. Image of R.V.: all possible values X can take Stat 330 (Spring 2015): slide set 7 ⎧ ⎨ −x 2x h(x) = ⎩ 0 for x = 1, 3, 5 for x = 2, 4 for x = 6. 10 For that, we look at another function, h(x), that counts the money I win with respect to the number of spots: of of of of of of all all all all all all will will will will will will be be be be be be 1, 2, 3, 4, 5, 6, and and and and and and I I I I I I will will will will will will gain gain gain gain gain gain -1 4 -3 8 -5 0 dollars dollars dollars dollars dollars dollars · (−1) + 16 · 4 + 16 · (−3) + 16 · 8 + 16 · (−5) + 16 · 0 = 3 6 = 0.5 11 We denote this by E(h(X)) and deﬁne this mathematically in the next lecture. In this example, we are calculating the expected value of the function h(X) dollars per play. 1 6 X X X X X X Gambling Example Continued... tosses tosses tosses tosses tosses tosses In total I expect to get 1/6 1/6 1/6 1/6 1/6 1/6 Stat 330 (Spring 2015): slide set 7 Stat 330 (Spring 2015): slide set 7 In In In In In In 9 1 2 3 4 1/6 1/3 1/3 1/6 The diagram shows all six faces of a particular die. If Z denotes the number of spots on the upturned face after toss this die, what is the probability mass function for Z? Assuming, that each face of the die appears with the same probability, we have 1 possibility to get a 1 or a 4, and two possibilities for a 2 or 3 to appear, which gives a probability mass function for Z as: Example: Roll of a doctored die z p(z) Stat 330 (Spring 2015): slide set 7 More Examples of Discrete R.V.’s 8 be the number of spots turned up, then if I pay you $X you pay me $2X no money changes hands. What amount of money do I expect to win? Gamblers Luck!: Toss a die. Let X ⎧ ⎨ 1, 3 or 5 2 or 4 X= ⎩ 6 Statistics of R.V.s The probability mass function for Y therefore is x 1 2 3 4 5 6 pX (x) 16 16 61 61 61 16 Assuming, that the die is a fair die means, that the outcomes of each face turning up are equally likely i.e., probabilities of each face turning up are equal. Obviously, Y is a random variable with image Im(Y ) = {1, 2, 3, 4, 5, 6}. Let Y be the number of spots on the upturned face of a die. Example: Roll of a fair die More Examples of Discrete R.V.’s