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1 Iowa State University Feb 21, 2013 Iowa State University Feb 21, / 31
2 Outline Iowa State University Feb 21, / 31
3 random variables Two types of random variables: Discrete random variable: one that can only take on a set of isolated points (X, N, and S). random variable: one that can fall in an interval of real numbers (T and Z). Examples of continuous random variables: Z the amount of torque required to loosen the next bolt (not rounded). T the time you ll have to wait for the next bus home. C outdoor temperature at 3:17 PM tomorrow. L length of the next manufactured part. Iowa State University Feb 21, / 31
4 random variables V : % yield of the next run of a chemical process. Y : % yield of a better process. How do we mathematically distinguish between V and Y, given: Each has the same range: 0% V, Y 100% There are uncountably many possible values in this range. We want to show that Y tends to take on higher % yield values than V. Iowa State University Feb 21, / 31
5 V and Y have continuous probability s of V of Y f(v) f(y) v y The heights of these curves are not themselves probabilities. However, the the curves tell us that process Y will yield more product per run on average than process X. Iowa State University Feb 21, / 31
6 A generic probability density function (pdf) Iowa State University Feb 21, / 31
7 Outline Iowa State University Feb 21, / 31
8 Definition: probability density function (pdf) A probability density function (pdf) of a continuous random variable X is a function f (x) with: f (x) 0 for all x. f (x)dx 1 P(a X b) b a f (x)dx, a b The pdf is the continuous analogue of a discrete random variable s probability mass function. Iowa State University Feb 21, / 31
9 Example Let Y be the time delay (s) between a 60 Hz AC circuit and the movement of a motor on a different circuit. Say Y has a density of the form: { c 0 < y < 1 f (y) 60 0 otherwise we say that Y has a Uniform(0, 1/60). f (y) must integrate to 1: 1 f (y)dy 0 0dy + 1/60 0 cdy + 0dy c 1/60 60 hence, c 60, and: f (y) { 60 0 < y < otherwise Iowa State University Feb 21, / 31
10 A look at the density Iowa State University Feb 21, / 31
11 Your turn: calculate the following probabilities. f (y) 1. P(Y ) 2. P(Y > 1 70 ) 3. P( Y < ) 4. P( Y 1 > P(Y 1 80 ) 110 ) { 60 0 y otherwise Iowa State University Feb 21, / 31
12 Answers: calculate the following probabilities 1. P(Y ) P( < Y ) 1/100 0 f (y)dy 0dy / dy Iowa State University Feb 21, / 31
13 2. P(Y > 1 70 ) P( 1 70 < Y ) 1/70 1/60 1/70 f (y)dy 60dy + 0dy 1/60 60y 1/60 1/70 ( ) Iowa State University Feb 21, / 31
14 3. P( Y < ) P( < Y < ) 1/120 1/ /120 f (y)dy 0dy + 1/ y 1/120 0 ( ) dy Iowa State University Feb 21, / 31
15 4. P( Y > ) P(Y > or Y < ) P(Y > or Y < ) P(Y > ) + P(Y < ) 31/2200 1/60 31/2200 f (y)dy + 60dy + 9/2200 1/60 0dy + f (y)dy 9/ /60 31/2200 ( ) dy Iowa State University Feb 21, / 31
16 5. P(Y 1 80 ) P( 1 80 Y 1 80 ) 1/80 1/80 f (y)dy 1/80 1/80 ( 60 1/80 1 1/ ) dy In fact, for any random variable X and any real number a: P(X a) P(a X a) a a f (x)dx 0 Iowa State University Feb 21, / 31
17 Outline Iowa State University Feb 21, / 31
18 functions (cdf) The cumulative function of a random variable X is a function F such that: In other words: F (x) P(X x) d F (x) f (x) dx x f (t)dt As with discrete random variables, F has the following properties: F (x) 0 for all x. F is monotonically increasing. lim x F (x) 0 lim x F (x) 1 Iowa State University Feb 21, / 31
19 Example: calculating the cdf of Y Remember: For y 0: For 0 < y < 1/60: f Y (y) { 60 0 < y < 1/60 0 otherwise y 0 F (y) P(Y y) f (t)dt 0dt 0 y 0 y F (y) P(Y y) f (t)dt 0dt + 60dt 60y 0 For y 1/60: y F (y) P(Y y) 0 0dt + f (t)dt 1/ dt + 0dt 1 1/60 Iowa State University Feb 21, / 31
20 A look at the cdf Iowa State University Feb 21, / 31
21 Your turn: calculate the following using the cdf 1. F (1/70) 2. P(Y 1 80 ) 3. P(Y > ) 4. P( Y ) 0 y 0 F (y) 60y 0 < y 1 1 y > Iowa State University Feb 21, / 31
22 Answers: calculate the following using the cdf 1. F ( 1 70 ) P(Y 1 80 ) F ( 1 80 ) P(Y > ) 1/150 f (y)dy f (y)dy 1/150 1 F (1/150) f (y)dy 3 5 In fact, for any random variable X, discrete or continuous: P(X x) 1 P(X < x) Iowa State University Feb 21, / 31
23 4. P( Y ) 1/120 1/130 1/120 f (y)dy f (y)dy 1/130 F (1/120) F (1/130) 60(1/120) 60(1/130) 1/ f (y)dy Iowa State University Feb 21, / 31
24 Outline Iowa State University Feb 21, / 31
25 The A random variable X has an Exponential(α) if: { 1 f (x) α e x/α x > 0 0 otherwise Iowa State University Feb 21, / 31
26 Your turn: for X Exp(2), calculate the following f (x) 1. P(X 1) 2. P(X > 5) 3. The cdf F of X { 1 2 e x/2 x > 0 0 otherwise Iowa State University Feb 21, / 31
27 Answers: for X Exp(2), calculate the following 1. P(X 1) 1 0 f (x)dx 1 1 0dx e x/2 dx 0 + ( e x/2 1 0 e 1/2 ( e 0/2 ) 1 e 1/ Iowa State University Feb 21, / 31
28 2. P(X > 5) 5 5 f (x)dx e x/ e x/2 dx e /2 + e 5/2 e 5/ Iowa State University Feb 21, / 31
29 3. For x < 0: For x 0: F (x) P(X x) x 0dx 0 F (x) P(X x) 0 0dx + x x x 0 f (x)dx) f (x)dx 1 2 e t/2 dt e t/2 x 0 e x/2 ( e 0/2 ) 1 e x/2 Iowa State University Feb 21, / 31
30 Hence: F (x) { 1 e x/2 x 0 0 otherwise In general, an Exp(α) random variable has cdf: { 1 e x/α x 0 F (x) 0 otherwise Iowa State University Feb 21, / 31
31 Uses of the Exp(α) random variable An Exp(α) random variable measures the waiting time until a specific event that has an equal chance of happening at any point in time. Examples: Time between your arrival at a bus stop and the moment the bus comes. Time until the next person walks inside the library. Time until the next car accident on a stretch of highway. Iowa State University Feb 21, / 31
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