Download Summary of Chapter Three Random Variables: Let S be the sample

Summary of Chapter Three
Random Variables: Let S be the sample space of an experiment. A real-valued function X : S → R is called a random
variable.If X takes finite or countable number of possible values, then X is said to be discrete.
Distribution Function: For a random variable X, its distribution function F (x) is defined by F (x) = P (X ≤ x).
Probability Mass Functions, Expectation and Variance:
Let X be a discrete random variable with a space S = {x1, x2, · · · }
and a probability mass function f (x) = P (X = x), then
(1)
X
f (x) = 1
x∈S
(2)
F (x) =
X
f (x)
y≤x,y∈S
(3)
µ = E(X) =
X
xf (x),
x∈S
2
E(X ) =
X
x2f (x)
x∈S
(4)
σ 2 = V (X) = E[(X − µ)2] = E(X 2) − µ2,
p
σ = V (X)(standard deviation).
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(5)
E[h(X)] =
X
h(x)f (x).
x∈S
Special Discrete Random Variables:
(i)Let X be the number of successes in n Bernoulli trials (with
probability p for a success in each trial). Then, X is a binomial
random variable with parameters (n, p). Its probability mass
function is given by
n x
f (x) = P (X = x) =
p (1 − p)n−x , x = 0, 1, · · · , n
x
Its mean (or expected value) and variance are given by
E(X) = np,
V (X) = np(1 − p)
(ii)Let X be the number of trials needed to observe the first success. Then X is a geometric random variable with parameter
p. Its probability mass function is given by
f (x) = p(1 − p)x−1, x = 1, 2, · · ·
Its mean and variance are given by
1
1−p
E(X) = , V (X) = 2 .
p
p
(iii) Let X be a hypergeometric random variable with parameters n, N, K. Its probability mass function is given by
K N −K
f (x) =
x
n−x
N
n
, x = max{0, n + K − N }to min{K, n},
2
With p = K/N , its mean and variance are given by
E(X) = np,
V (X) =
N −n
np(1 − p)
N −1
(v) Let X be a Poisson random variable with parameter λ. Its
probability mass function is given by
e−λλx
f (x) =
, x = 0, 1, 2, · · ·
x!
Its mean (or expected value) and variance are given by
E(X) = V (X) = λ.
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