Consider a probability space (S, C, P). RANDOM VARIABLES
41 Consider a probability space (S, C, P). RANDOM VARIABLES
DEFINITION 41.1. A random variable X on the sample space S is a function from S into the set R of real numbers such that the preimage of every interval of R is an event of S. If S is a discrete sample space in which every subset of S is an event, then every real-valued function on S is a random variable. On the other hand, if S is uncountable, then certain real-valued functions on S may not be random variables.
Let X be a random variable on S, where we let R X denote the range of X; that is,
R X = {x | there exists s ∈ S for which X(s) = x}
There are two cases that we treat separately. (i) X is a discrete random variable; that is, R X is finite or countable. (ii) X is a continuous random variable; that is, R X is a continuum of numbers such as an interval or a union of intervals. Let X and Y be random variables on the same sample space S. Then, as usual, X + Y, X + k, kX, and XY (where k is a real number) are the functions on S defined as follows (where s is any point in S):
(X + Y)(s) = X(s) + Y(s), (kX)(s) = kX(s), (X + k)(s) = X(s) + k, (XY)(s) = X(s)Y(s).
More generally, for any polynomial, exponential, or continuous function h(t), we define h(X) to be the function on S defined by
[h(X)](s) = h[X(s)]
One can show that these are also random variables on S. The following short notation is used:
P(X =x i )
denotes the probability that X =x i .
P(a ≤X≤b denotes the probability that X lies in the closed interval [a, b]. µ X or E(X) or simply µ
denotes the mean or expectation of X. σ X 2 or Var(X) or simply 2 σ denotes the variance of X.
σ X or simply σ
denotes the standard deviation of X.
Sometimes we let Y
be a random variable such that Y = g(X), that is, where Y is some function of X.
Discrete Random Variables Here X is a random variable with only a finite or countable number of values, say
R X = {x 1 ,x 2 ,x 3 , …}where, say, x 1 <x 2 ,<x 3 < …. Then X induces a function f(x) on R X as follows:
f(x i ) = P(X = x i ) = P({s ∈ S | X(s) = x i })
The function f(x) has the following properties:
(i) f(x i ) ≥ 0 and (ii) Σ i f(x i ) =1
Thus, f defines a probability function on the range R X of X. The pair (x i , f(x i )), usually given by a table, is called the probability distribution or probability mass function of X.
RANDOM VARIABLES
Mean
41.1. µ X = E(X) = Σx i f(x i ) Here, Y = g(X).
41.2. µ Y = E(Y) = Σg(x i ) f(x i )
Variance and Standard Deviation
41.3. σ X 2 = Var(X) = Σ(x
i – m) f(x i ) = E((X – m) )
Alternately, Var(X) =s 2 may be obtained as follows:
41.4. Var(X) = Sx 2 f(x
i )–m = E(X )–m
41.5. σ X = Var X () =EX ( 2 ) − µ 2
REMARK: Both the variance Var(X) =s 2 and the standard deviation s measure the weighted spread of the values x i about the mean m; however, the standard deviation has the same units as m.
EXAMPLE 41.1: Suppose X has the following probability distribution:
0.1 0.2 0.3 0.4 Then:
f(x)
m = E(X) = Σx i f(x i ) = 2(0.1) + 4(0.2) + 6(0.3) + 8(0.4) = 6 E(X 2 ) = Σx 2 f(x i ) =2 2 (0.1) 2 (0.2) +6 2 (0.3) +8 i 2 +4 (0.4) = 40 s 2 = Var(X) = E(X 2 ) −m 2 = 40 − 36 = 4
s = Var X ()=4=2
Continuous Random Variable Here X is a random variable with a continuum number of values. Then X determines a function f(x), called
the density function of X, such that
(i) f(x) ∞ ≥ 0 and (ii) f(x) dx
= ∫ R f x dx =1
Furthermore,
P(a b ≤ X ≤ b) = ∫
a f(x) dx
Mean
41.6. μ X = E(X) =
∫ xf(x) dx
Here, Y = g(X).
41.7. μ Y = E(Y) =
∫ g(x) f(x) dx
RANDOM VARIABLES
Variance and Standard Deviation
41.8. σ X 2 = Var(X) = ∞ (x
− m) 2 f(x)dx = E((X − m) 2 ∫ )
Alternately, Var(X) =s 2 may be obtained as follows:
41.9. Var(X) =
x 2 f(x)dx −m 2 = E(X 2 ) −m ∫ 2
41.10. s X = Var X () =EX ( 2 ) − μ 2
EXAMPLE 41.2: Let X be the continuous random variable with the following density function:
xf(x) dx
x f(x) dx
x 3 dx
s 2 = Var(X) = E(X 2 ) −m 2 =2−
s = Var X ()= 9= 32
Cumulative Distribution Function The cumulative distribution function F(x) of a random variable X is the function F:R → R defined by
41.11. F(a) = P(X ≤ a) The function F is well-defined since the inverse of the interval ( −∞, a] is an event. The function F(x) has the following properties:
41.12. F(a) ≤ F(b) whenever a ≤ b.
41.13. x lim →−∞ F(x) = 0 and lim x →+∞ F(x) =1 That is, F(x) is monotonic, and the limit of F to the left is 0 and to the right is 1.
If X is the discrete random variable with distribution f(x), then F(x) is the following step function:
41.14. F(x) = f(x
If X is a continuous random variable, then the density funcion f(x) of X can be obtained from the cum- mulative distribution function F(x) by differentiation. That is,
= F′(x) Accordingly, for a continuous random variable X,
f(x) 41.15. d = F(x) dx
41.16. x F(x) = f(t) dt ∫
RANDOM VARIABLES
Standardized Random Variable The standardized random variable Z of a random variable X with mean m and standard deviation s > 0 is
defined by
41.17. Z X = − µ σ Properties of such a standardized random variable Z follow:
µ Z = E(Z) = 0 and σ Z =1
EXAMPLE 41.3: Consider the random variable X in Example 41.1 where µ X = 6 and σ X = 2. The distribution of Z = (X – 6)/2 where f(z) = f(x) follows:
0 1 f(Z)
E(Z) =Σz i f(z i ) = (−2)(0.1) + (−1)(0.2) + 0(0.3) + 1(0.4) = 0 E(Z 2 ) =Σz 2 i f(z i ) = (−2) 2 (0.1) + (−1) 2 (0.2) +0 2 (0.3) +1 2 (0.4) =1
Var(Z) =1−0 2 = 1 and s Z = Var X () =1
Probability Distributions
Φ(x) = ∑ p t q n-t p > 0, q > 0, p t ≤ x
41.18. Binomial Distribution:
41.19. λ Poisson λ Distribution: Φ(x) = t ∑
z ⎝⎜ ⎠⎟ ⎝⎜ − t ∑ ≤ x ⎛ r + s ⎞
41.20. Hypergeometric Distribution:
Φ(x) =
⎝⎜ n ⎠⎟
41.21. 2 Normal Distribution: e − t / Φ(x) = 2
dt
2 ⎠⎟ x ⎛ t Student’s 2 Distribution: 1 41.22. ⎞ t
−+ ( n 12 1 )/ ⎝⎜
Φ(x) =
n π Γ (/) n 2 ∫ ⎝⎜ n ⎠⎟
dt d
41.23. x c 2 1
(Chi Square) Distribution: Φ(x) =
2 Γ (/) n 2 ∫ 0
(n - 2)/2
e -t/2 dt
− ( n + n )/ 41.24. 2 F Distribution: t 1 1 Φ(x) = 2 ( n + nt ) dt
(( / ) n 21 −
Γ (/)(/) n 1 2 n
Section XII: Numerical Methods