Survey

Survey

Document related concepts

no text concepts found

Transcript

Lecture 5 - Random variables Learning Objectives • After completing this lecture, the students should be able to define and work with – Random variables (Discrete and Continuous). – Probability mass functions and density functions. – Cumulative distribution functions. – Expectations and variances of random variables. STAT 101 1 Random variables Random variable Let S be a sample space corresponding to the random experiment E. A function X defined on the sample space S to the real number system R is called a random variable. R S s X(s)=x • Note that random variables are written by uppercase letters. • The values taken by the random variables are written by the corresponding lowercase letters.. STAT 101 2 Random variables Example 5.1 • E – Tossing a fair coin twice and let X be the number of heads obtained. • The sample space S = {TT, TH, HT, HH} X (TT ) 0, X (TH ) X ( HT ) 1 and X ( HH ) 2 • The Set R = {0,1, 2} is called the range space of X. Discrete Random Variable • Assumes a finite or countably infinite number of values. • The values can be enumerated and listed as x1 , x2 ,....., xn ,.... STAT 101 3 Random variables Continuous Random variables • Assumes uncountable number of values from a region. Examples • X= Height of an individual in our class (Continuous random variable). • X = Number of Female students in our class ( Discrete random variable) • X = Number of tosses required of a coin to obtain Head for the fifth time. (Discrete random variable) STAT 101 4 Random variables Probability mass function • Let X be a discrete random variable with range space R {x1 , x2 ,......, xn ,....}. • With each value of X we assign a nonnegative real number satisfying the following conditions ( a ) p ( xi ) 0 and x1 x2 ........xn ......... p x1 p x2 ... p xn ....... (b) p ( xi ) 1 1 • The assignment xi , p( xi ) i 1, 2,...n is called the probability mass function of X. STAT 101 5 Lecture 4 - Random variables Example 5.2 • Let X be the number of heads obtained. The range space of X is RX 0, 1, 2. • We assign the a nonnegative mass with each value as shown P( X 0) 1 4 P( X 1) 2 4 P( X 2) 1 4 • The assignment is a valid probability assignment and hence is known as probability mass function of X. P(x) 1/2 1/4 0 1 2 STAT 101 x 6 Random variables • To find the probability of an event A, we add the probabilities attached to the elements of the event. P( A) P( x) xA • For example, P( X 1) P( X 1) P( X 2) 1 1 3 2 4 4 STAT 101 7 Random variables Probability density function (pdf) A function f (x) can be a probability density function of a continuous random variable X if it satisfies the following conditions (a ) f ( x) 0 for all x (b) f ( x) dx 1 Note that f ( x ) at any specific point x0 has no meaning. • To find the probability of an event, we integrate the density function over the event. That is, 0 b P(a X b) f ( x) dx a STAT 101 8 Random variables Example 5.3 Consider the function 2 x, 0 x 1 f ( x) 0, elsewhere Since, 0 x 1, f ( x) 0 for all x. 1 0 1 x2 f ( x) dx 2 x dx 2 2 0 1 0 1 0 1 The function To find f (x ) is a valid pdf. 1 x2 P (0.5 X 1) 2 x dx 2 2 0.5 1 1 0.5 STAT 101 1 3 4 4 9 Random variables Cumulative Distribution Function (CDF) The cumulative distribution function of a random variable X is defined as F ( x) P( X x) • For a discrete random variable F ( x) P( x) x X x X x • For a continuous random variable x F ( x) f ( x)dx STAT 101 10 Random variables Example 5.4 • Consider the prob. distribution in example 4.2. P( X 0) 1 4 P( X 1) 2 4 P( X 2) 1 4 P(x) 1/2 F (x) 1 1/4 p ( 2) 3/4 p (1) 0 1 2 1/4 x p (0) 0 0 if x 0 0.25 if 0 x 1 F ( x) 0.75 if 1 x 2 if x 1 1 • The CDF is STAT 101 1 x 2 Properties of F(x) ? 11 Random variables Example 5.5 Consider the density function f (x) 0 1 2 x, 0 x 1 f ( x) 0, elsewhere x The cumulative distribution function F (x) 0 if x 0 x F ( x) 2 x dx x 2 if 0 x 1 0 1 if x 1 1 x2 0 STAT 101 1 12 Random variables Properties (a) The CDF of a continuous random variable is a continuous function. (b) Non decreasing function (c) F () 0 and F () 1 d F ( x ) (d) The pdf f ( x) dx STAT 101 13 Random variables Characteristics of a random variables Expected value of a random variable Let X be a discrete random variable with known probability distribution p(x). The expected value of X is defined as E ( X ) xi p ( xi ) 1 • E(X) is a weighted average of all possible values of X. • E(X) is a population mean and we denote it by X • For a continuous random variable with pdf f(x), E( X ) x f ( x) dx STAT 101 14 Random variables Properties (a) E(c) = c, where c is a constant. (b) E(a + b X) = a + b E(X), where a and b are constants. (c) E (X + Y ) = E (X) + E ( Y) (d) E ( XY ) = E(X) E(Y) only if X and Y are independent (discussed in lecture 6) (e) Let X be a random variable and let Y = g(X), then E (Y ) g ( xi ) p ( xi ) (discrete ) 1 STAT 101 E (Y ) g ( x) f ( x) dx 15 Random variables Example 5.6 The probability distribution of daily demand for a product is d p(d) 1 2 3 4 5 0.1 0.1 0.3 0.3 0.2 Evaluate E(D). Solution BY definition 5 E ( D) d i p(d i ) 1 1(0.1) 2(0.1) 3(0.3) 4(0.3) 5(0.2) 0.1 0.2 0.9 1.2 1.0 3.4 STAT 101 16 Random variables Variance • measures the spread of values of X around its mean value. • measured in the squared units of measurement for X. • defined as V ( X ) X2 E{ X E ( X )}2 • The positive square root of V(X) is called the standard deviation of X and is denoted by X • Variance X2 E{ X E ( X )}2 E{ X 2 E X 2 XE ( X ) 2 E X 2 E X 2E X 2 2 V ( X ) E X 2 E X STAT 101 2 17 Random variables Properties (a) V (c) = 0, where c is a constant. (b) V(X + c) = V(X) 2 (c) V(c X) = c V ( X ) (d) If X and Y are independent, then V(X + Y) = V(X) + V(Y) STAT 101 18 Random variables Example 5.7 24 Suppose that X has pdf f ( x) 4 , x 2. Evaluate the variance x of X. Solution By definition E( X ) x f ( x) dx 2 x 2 24 1 3 dx 24 x dx 24 2 x4 2x2 STAT 101 3 2 19 Random variables Example 5.7 E( X 2 ) 2 x f ( x) dx 2 x2 2 24 1 2 dx 24 x dx 24 2 x4 x 12 2 V ( X ) E X 2 EX 2 12 32 3 STAT 101 20

Related documents