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Chapt 2b

Chapt 2b - 2.5 Discrete Random Variables 69 Thus given the...

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2.5 Discrete Random Variables 69 Thus, given the plot of the CDF of a discrete random variable, we can obtain the PMF of the random variable by noting that the random variable only takes on values that have nonzero probability at those points where jumps occur. The probability that the random variable takes on any other value than where the jumps occur is zero. Furthermore, the probability that the random variable takes a value where a jump occurs is equal to the size of the jump. Example 2.9 The plot of the CDF of a discrete random variable X is shown in Figure 2.7. Find the PMF of X . Solution The random variable takes on values with nonzero probability at X = 1, X = 2, X = 4, and X = 6. The size of the jump at X = 1 is 1 / 3, the size of the jump at X = 2 is 1 / 2 - 1 / 3 = 1 / 6, the size of the jump at X = 4 is 3 / 4 - 1 / 2 = 1 / 4, and the size of the jump at X = 6 is 1 - 3 / 4 = 1 / 4. Thus, the PMF of X is given by p X ( x ) = 1 / 3 x = 1 1 / 6 x = 2 1 / 4 x = 4 1 / 4 x = 6 0 otherwise trianglesolid Figure 2.7 Graph of F X ( x ) for Example 2.9

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70 Chapter 2 Random Variables Example 2.10 Find the PMF of a discrete random variable X whose CDF is given by F X ( x ) = 0 x < 0 1 / 6 0 x < 2 1 / 2 2 x < 4 5 / 8 4 x < 6 1 x 6 Solution In this example, we do not need to plot the CDF. We observe that it changes values at X = 0, X = 2, X = 4, and X = 6, which means that these are the values of the random variable that have nonzero probabilities. The next task after isolating these values with nonzero probabilities is to determine their probabilities. The first value is p X ( 0 ) , which is 1 / 6. At X = 2, the size of the jump is 1 / 2 - 1 / 6 = 1 / 3 = p X ( 2 ) . Similarly, at X = 4, the size of the jump is 5 / 8 - 1 / 2 = 1 / 8 = p X ( 4 ) . Finally, at X = 6, the size of the jump, which is the value of p X ( 6 ) , is 1 - 5 / 8 = 3 / 8. Therefore, the PMF of X is given by p X ( x ) = 1 / 6 x = 0 1 / 3 x = 2 1 / 8 x = 4 3 / 8 x = 6 0 otherwise trianglesolid 2.6 Continuous Random Variables Discrete random variables have a set of possible values that are either finite or countably infinite. However, there exists another group of random variables that can assume an uncountable set of possible values. Such random variables are called continuous random variables. Thus, we define a random variable X to be a continuous random variable if there exists a nonnegative function f X ( x ) , defined for all real x ( -∞ , ) , having the property that for any set A of real numbers, P ( X A ) = integraldisplay A f X ( x ) dx The function f X ( x ) is called the probability density function (PDF) of the random variable X and is defined by f X ( x ) = dF X ( x ) dx The properties of f X ( x ) are as follows:
2.6 Continuous Random Variables 71 1. f X ( x ) 0 2. Since X must assume some value, integraltext -∞ f X ( x ) dx = 1 3. P [ a X b ] = integraltext b a f X ( x ) dx , which means that P [ X = a ] = integraltext a a f X ( x ) dx = 0.

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Chapt 2b - 2.5 Discrete Random Variables 69 Thus given the...

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