<p> 1</p><p>BA 1605</p><p>Chapter 7 Random Variables and Discrete Probability Distributions</p><p>In this chapter we combine the concepts of descriptive statistics (Chapter 4) and probability (Chapter 6) when learning about random variables and their probability distributions.</p><p>7.1 Random Variables and Probability Distributions</p><p> A random variable (usually denoted as X) is a variable whose outcome is determined by chance.  A discrete random variable has a finite number of values or a countably infinite number of values. For example: X = number of days that it rained in May, or X= number of heads in three tosses of a coin, or X = number of flaws in two metres of fabric.  A continuous random variable has infinitely many values and is associated with measurements on a continuous scale without gaps or interruptions. For example: X = weight of 10 year old boys, X = diameter of bolts  A probability distribution is a description that gives the probability for each value of the random variable. It is often expressed as a table, a graph or a formula. We will only examine discrete probability distributions in this chapter</p><p>Example: In an experiment that consists of flipping a coin three times, let X be the number of heads observed. Note that X can take on the values 0, 1, 2, 3. The probability distribution of X can be expressed in the following ways:</p><p>0 0.125 1 0.375 2 0.375 3 0.125 2</p><p>Probability Distribution of X</p><p>0.4 x</p><p> f 0.3 o</p><p> y t i l</p><p> i 0.2 b a b o</p><p> r 0.1 p</p><p>0 Number of heads in three tosses</p><p>3 n 骣1 P( X= x ) = C 琪 Note: This is an example of a binomial probability x 桫2 distribution that we will learn about in the next section.</p><p>Properties of Discrete Probability Distributions</p><p>1. 0＃P ( x ) 1 , for every value of x = 2. P( x ) 1 x</p><p>Example:</p><p>Determine whether the following are valid probability distributions: 1 a) P( x) =, x = 1,2,3,4 yes 4 1 P( x) =, x = 0,1,2 b) 2( 2- x) ! x ! yes 1 c) P( x) =, x = 1,2,3 no x</p><p>Mean of the Random Variable X</p><p> Random variables also have mean, variance and standard deviations. We use the concept of expected value to calculate them.  The mean of a discrete random variable is the long term average if we had repeated the experiment an infinite number of times. It is the expected value after an infinite number of repetitions. 3</p><p> The mean or expected value of X, denoted  or E(X), can be found by E X  xP x the following formula, in the discrete case:     . In the x continuous case, we use calculus and the concept of integration instead of summation, but not in this class.</p><p>Example </p><p>X= # of heads in three tosses 骣1 骣 3 骣 3 骣 1 12 E( X ) =0琪 + 1 琪 + 2 琪 + 3 琪 = = 1.5 桫8 桫 8 桫 8 桫 8 8 Note that the E(X) does not have to take on a possible value of X, since it is the long term average.</p><p>The variance and standard deviation of X is more complicated than that of the mean. Before starting, let us look at some properties of expected value.</p><p>= Definition: E[ g( x)] g( x) P( x ) x</p><p>Example:</p><p>+ = +( ) = + = + a) E(3 X 5) 邋( 3 x 5) P x[ 3 xP( x) 5 P( x)] 邋 3 xP( x) 5 P( x ) x x x x E(3 X+ 5) = 3邋 xP( x) + 5 P( x) = 3 E( X ) + 5 x x = = = b) E(3) 邋 3 P ( x ) 3 P( x ) 3 x x Properties of Expected Value</p><p>Where a, b, c are constants and X is a random variable</p><p>1. E(c) = c 2. E( ax+ b) = aE( x) + b</p><p>Variance of Discrete Random Variable X</p><p>2 2 2 Var (X) or s=E( X - m) =( x - m ) P( x ) x 4</p><p>Example </p><p>X= # of heads in three tosses E( X ) = 1.5 2 2 s 2 =E( X -1.5) =( x - 1.5) P( x ) x</p><p>2骣1 2 骣 3 2 骣 3 2 骣 1 6 3 s 2 =(0 - 1.5) 琪 +( 1 - 1.5) 琪 +( 2 - 1.5) 琪 +( 3 - 1.5) 琪 = = 桫8 桫 8 桫 8 桫 8 8 4</p><p>Shortcut method for the Variance of Random Variable X</p><p>2 2 2 2 2 2 2 2 s=E( X ) - m or s=E( X) -臌轾 E( X) = x P( x ) - m x</p><p>Proof: 2 2 s2 =E( X - m) =( x - m ) P( x ) x s2=(x 2 -2 m x + m 2 ) P( x ) x s2=邋x 2 P( x) -2 m xP( x) + m 2 P( x ) x x x s2=E( X 2) - m 2</p><p>Example </p><p>X= # of heads in three tosses E( X ) = 1.5 骣1 骣 3 骣 3 骣 1 24 E( X2) = x 2 P( x ) =0 2琪 + 1 2 琪 + 2 2 琪 + 3 2 琪 = = 3 x 桫8 桫 8 桫 8 桫 8 8 s2=E( X 2) - m 2 =3 - 1.5 2 = 0.75</p><p>Standard Deviation for Discrete Random Variable X</p><p>2 22 2 2 SD(X) or s=(x - m ) P( x ) or s=E( X) -臌轾 E( X) = x P( x ) - m x x 5</p><p>Example </p><p>X= # of heads in three tosses 3 3 s = = 0.866 4 2</p><p>Properties of Variance</p><p>1. V( c ) = 0 2. V( X+ c) = V( X ) 3. V( aX+ c) = a2 V( X )</p><p>Exercises page 217 - 221</p>