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Table 2-1 Sample Point X HH 2 HT 1 TH 1 TT 0
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It should be noted that many other random variables could also be defined on this sample space, for example, the square of the number of heads or the number of heads minus the number of tails.
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A random variable that takes on a finite or countably infinite number of values (see page 4) is called a discrete random variable while one which takes on a noncountably infinite number of values is called a nondiscrete random variable.
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Discrete Probability Distributions
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Let X be a discrete random variable, and suppose that the possible values that it can assume are given by x1, x2, x3, . . . , arranged in some order. Suppose also that these values are assumed with probabilities given by P(X xk) f(xk) k 1, 2, . . . (1)
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It is convenient to introduce the probability function, also referred to as probability distribution, given by P(X x) f(x) 0. (2)
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For x xk, this reduces to (1) while for other values of x, f(x) In general, f (x) is a probability function if 1. f (x)
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2. a f (x)
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where the sum in 2 is taken over all possible values of x.
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CHAPTER 2 Random Variables and Probability Distributions
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EXAMPLE 2.2 Find the probability function corresponding to the random variable X of Example 2.1. Assuming that the coin is fair, we have
P(HH ) Then P(X P(X P(X 0) 1) 2)
P(HT )
P(TH )
P(T T )
P(T T)
1 4 P(HT ) P(TH ) 1 4 1 4 1 2
P(HT < TH ) P(HH) 1 4
The probability function is thus given by Table 2-2.
Table 2-2
x f (x) 0 1> 4 1 1> 2 2
Distribution Functions for Random Variables
The cumulative distribution function, or briefly the distribution function, for a random variable X is defined by F(x) P(X x) (3) where x is any real number, i.e., ` x ` . The distribution function F(x) has the following properties: 1. F(x) is nondecreasing [i.e., F(x) 2. lim F(x) 0; lim F(x) 1. S S
x ` x `
F( y) if x
y]. h) F(x) for all x].
3. F(x) is continuous from the right [i.e., lim F(x
Distribution Functions for Discrete Random Variables
The distribution function for a discrete random variable X can be obtained from its probability function by noting that, for all x in ( ` , ` ), F(x) P(X x) a f (u)
where the sum is taken over all values u taken on by X for which u x. If X takes on only a finite number of values x1, x2, . . . , xn, then the distribution function is given by 0 f (x1) e f (x1) ` x1 x2 f (xn) xn x x x ( x x1 x2 x3 `
F(x)
f (x2) c
( f (x1)
EXAMPLE 2.3
(a) Find the distribution function for the random variable X of Example 2.2. (b) Obtain its graph.
(a) The distribution function is
1 d4 3 4
F(x)
` 0 1 2
x x x x
0 1 2 `
CHAPTER 2 Random Variables and Probability Distributions
(b) The graph of F(x) is shown in Fig. 2-1.
Fig. 2-1
The following things about the above distribution function, which are true in general, should be noted. 1. The magnitudes of the jumps at 0, 1, 2 are 1, 1, 1 which are precisely the probabilities in Table 2-2. This fact 4 2 4 enables one to obtain the probability function from the distribution function. 2. Because of the appearance of the graph of Fig. 2-1, it is often called a staircase function or step function. 1 3 The value of the function at an integer is obtained from the higher step; thus the value at 1 is 4 and not 4. This is expressed mathematically by stating that the distribution function is continuous from the right at 0, 1, 2. 3. As we proceed from left to right (i.e. going upstairs), the distribution function either remains the same or increases, taking on values from 0 to 1. Because of this, it is said to be a monotonically increasing function. It is clear from the above remarks and the properties of distribution functions that the probability function of a discrete random variable can be obtained from the distribution function by noting that f (x) F(x) lim F(u). (6)
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