1 / 61

690 likes | 1.91k Vues

Random Variables. an important concept in probability. A random variable , X, is a numerical quantity whose value is determined be a random experiment. Examples Two dice are rolled and X is the sum of the two upward faces.

Télécharger la présentation
## Random Variables

**An Image/Link below is provided (as is) to download presentation**
Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.
Content is provided to you AS IS for your information and personal use only.
Download presentation by click this link.
While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.
During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

**Random Variables**an important concept in probability**A random variable , X, is a numerical quantity whose value**is determined be a random experiment Examples • Two dice are rolled and X is the sum of the two upward faces. • A coin is tossed n = 3 times and X is the number of times that a head occurs. • We count the number of earthquakes, X, that occur in the San Francisco region from 2000 A. D, to 2050A. D. • Today the TSX composite index is 11,050.00, X is the value of the index in thirty days**Examples – R.V.’s - continued**• A point is selected at random from a square whose sides are of length 1. X is the distance of the point from the lower left hand corner. point X • A chord is selected at random from a circle. X is the length of the chord. chord X**Definition – The probability function, p(x), of a random**variable, X. For any random variable, X, and any real number, x, we define where {X = x} = the set of all outcomes (event) with X = x.**Definition – The cumulative distribution function, F(x),**of a random variable, X. For any random variable, X, and any real number, x, we define where {X≤x} = the set of all outcomes (event) with X ≤x.**Examples**• Two dice are rolled and X is the sum of the two upward faces. S , sample space is shown below with the value of X for each outcome**Graph**p(x) x**The cumulative distribution function, F(x)**For any random variable, X, and any real number, x, we define where {X≤x} = the set of all outcomes (event) with X ≤x. Note {X≤x} =f if x < 2. Thus F(x) = 0. {X≤x} ={(1,1)} if 2 ≤ x < 3. Thus F(x) = 1/36 {X≤x} ={(1,1) ,(1,2),(1,2)} if 3 ≤ x < 4. Thus F(x) = 3/36**Continuing we find**F(x) is a step function**A coin is tossed n = 3 times and X is the number of times**that a head occurs. The sample Space S = {HHH (3), HHT (2), HTH (2), THH (2), HTT (1), THT (1), TTH (1), TTT (0)} for each outcome X is shown in brackets**Graphprobability function**p(x) x**Examples – R.V.’s - continued**• A point is selected at random from a square whose sides are of length 1. X is the distance of the point from the lower left hand corner. point X • A chord is selected at random from a circle. X is the length of the chord. chord X**E**Examples – R.V.’s - continued • A point is selected at random from a square whose sides are of length 1. X is the distance of the point from the lower left hand corner. point X S An event, E, is any subset of the square, S. P[E] = (area of E)/(Area of S) = area of E**S**The probability function Thus p(x) = 0 for all values of x. The probability function for this example is not very informative**S**The Cumulative distribution function**The probability density function, f(x), of a continuous**random variable Suppose that X is a random variable. Let f(x) denote a function define for -∞ < x < ∞ with the following properties: • f(x) ≥ 0 Then f(x) is called the probability density function of X. The random, X, is called continuous.**Thus if X is a continuous random variable with probability**density function, f(x) then the cumulative distribution function of X is given by: Also because of the fundamental theorem of calculus.**Example**A point is selected at random from a square whose sides are of length 1. X is the distance of the point from the lower left hand corner. point X**Now**and**Recall**p(x) = P[X = x] = the probability function of X. This can be defined for any random variable X. For a continuous random variable p(x) = 0 for all values of X. Let SX ={x| p(x) > 0}. This set is countable (i. e. it can be put into a 1-1 correspondence with the integers} SX ={x| p(x) > 0}= {x1, x2, x3, x4, …} Thus let**Proof: (thatthe set SX ={x| p(x) > 0} is countable)**(i. e. can be put into a 1-1 correspondence with the integers} SX = S1 S2 S3 S3 … where i. e.**Thus the elements of SX = S1 S2 S3 S3 …**can be arranged {x1, x2, x3, x4, … } by choosing the first elements to be the elements of S1 , the next elements to be the elements of S2 , the next elements to be the elements of S3 , the next elements to be the elements of S4 , etc This allows us to write**A Discrete Random Variable**A random variable X is called discrete if That is all the probability is accounted for by values, x, such that p(x) > 0.**Discrete Random Variables**For a discrete random variable X the probability distribution is described by the probability function p(x), which has the following properties**Graph: Discrete Random Variable**p(x) b a**Continuousrandom variables**For a continuous random variable X the probability distribution is described by the probability density function f(x), which has the following properties : • f(x) ≥ 0**Graph: Continuous Random Variableprobability density**function, f(x)**A Probability distribution is similar to a distribution**ofmass. A Discrete distribution is similar to a pointdistribution ofmass. Positive amounts of mass are put at discrete points. p(x4) p(x2) p(x1) p(x3) x4 x1 x2 x3**A Continuous distribution is similar to a**continuousdistribution ofmass. The total mass of 1 is spread over a continuum. The mass assigned to any point is zero but has a non-zero density f(x)**The distribution function F(x)**This is defined for any random variable, X. F(x) = P[X ≤ x] Properties • F(-∞) = 0 and F(∞) = 1. Since {X ≤ - ∞} = f and {X ≤ ∞} = S then F(- ∞) = 0 and F(∞) = 1.**F(x) is non-decreasing (i. e. if x1 < x2 then F(x1) ≤F(x2)**) If x1 < x2 then {X ≤ x2} = {X ≤ x1} {x1 < X ≤ x2} Thus P[X ≤ x2] = P[X ≤ x1] + P[x1 < X ≤ x2] or F(x2) = F(x1) + P[x1 < X ≤ x2] Since P[x1 < X ≤ x2] ≥ 0 then F(x2) ≥F(x1). • F(b) – F(a) = P[a < X ≤ b]. If a < bthen using the argument above F(b) = F(a) + P[a < X ≤ b] Thus F(b) – F(a) = P[a < X ≤ b].**p(x) = P[X = x] =F(x) – F(x-)**Here • If p(x) = 0 for all x (i.e. X is continuous) then F(x) is continuous. A function F is continuous if One can show that Thus p(x) = 0 implies that**For Discrete Random Variables**F(x) is a non-decreasing step function with F(x) p(x)**For Continuous Random Variables Variables**F(x) is a non-decreasing continuous function with f(x) slope F(x) x**Success (S)**• Failure (F) Suppose that we have a experiment that has two outcomes These terms are used in reliability testing. Suppose that p is the probability of success (S) and q = 1 – p is the probability of failure (F) This experiment is sometimes called a Bernoulli Trial Let Then**The probability distribution with probability function**is called the Bernoulli distribution p q = 1- p

More Related