Download Random Variable A random variable is a function that assigns a

Random Variable
A random variable is a function that assigns a number to each point in a
sample space Ω.
Notation:
Random variable is denoted by upper case letters X, Y, Z, .... The realized
values of random variable is denoted by the lower case letter x = X(ω).
Properties
1) If X(ω), Y (ω) are random variables ( defined on the same sample space),
then X(ω) ± Y (ω), X(ω) · Y (ω), X(ω)
( provided Y (ω) 6= 0) are also random
Y (ω)
variables.
2) If f (x) is a function and X(ω) is a random variable, then f (X(ω)) is a
random variable as well.
If for any set A ⊂ R we know the probability
P (x ∈ A) = P {ω : X(ω) ∈ A}
we say that probability distribution of random variable X is given.
The distribution function of a random variable X is the function F : R −→
[0, 1] given by F (x) = P (X < x).
Let x and y be real numbers . A distribution function F (x) of a random
variable X satisfies the following properties
1) 0 ≤ F (x) ≤ 1 ∀x ∈ R
2) if x < y then F (x) ≤ F (y)
3) limx→−∞ F (x) = 0 and limx→∞ F (x) = 1
4) F is left-continuous
5) P (x ≤ X < y) = F (y) − F (x)
6) P (X = c) = limx→c+ F (x) − F (c)
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Discrete Random Variable
A discrete random variable is one which takes only a finite or countable
set of values. In this case a distribution function is a step function. If P (X =
P
xi ) = pi > 0 then xi is called a jump point. For jump points we have i pi = 1.
P
If A ⊂ R then P (X ∈ A) = xi ∈A pi .
If a random variable X has a discrete distribution given by (xi , pi ) then the
P
distribution function of X is F (x) = xi <x pi .
Example 1 A hunter has got 3 bullets and shoots until he succeds or runs out
of bullets. The random variable X describes the number of used bullets. Find
the probability distribution assuming that probability of hitting the target is 0.8.
Find the distribution function and P (1 < X ≤ 3), P (X > 0), P (X = 1.5).
Example 2 Consider a group of 4 boys and 6 girls. We choose 3 people at
random. Random variable X describes the number of chosen boys. Find the
probability distribution, the distribution function and P (X < 2), P (X ≥ 2).
Example 3 We are given distribution function of random variable X








F (x) = 






0 ; (−∞, −2 >
0, 2 ; (−2, 0 >
0, 4 ; (0, 2 >
0, 8 ; (2, 3 >
1 ; (3, ∞)
Find a probability distribution. Calculate
P (−2 < X < 3),
P (−2 < X ≤ 3),
P (−2 ≤ X < 3),
P (−2 ≤ X ≤ 3).
Example 4 Cosider random variable X such that P (X = k) =
1, 2, . . . . Find
a) C
b) F (x)
c) P (X > 5)
2
C
,
2k
k =
Example 5 Consider random variable X such that P (X = k) =
1, 2, . . . . Find
a) C
b) F (x)
c) P (X > 3)
3
C
,
k(k+1)
k=
Continuous Random Variable
R
X is continuous random variable if there exists nonnegative, integrable over
function f such that for each < x1 , x2 >
P (x1 ≤ X ≤ x2 ) =
Z x2
f (x) dx
x1
Function f is called a probability density function.
If f is a probability density function, then
a) fR (x) ≥ 0 for all x ∈ R
∞
b) −∞
f (x) dx = 1
If a random variable X has a probability density function f (x) then the
distribution function of X is
F (x) =
Z x
f (t) dt
−∞
In this case a distribution function is a continuous function.
Remark
P (X = x) = 0
Example 6 Consider function f (x) =
(
C(2x − x2 ) ; 0 ≤ x ≤ 2
0 ; elsewhere
Find
a) C
b) F (x)
c) P ( 21 ≤ X ≤ 1)
d) P (X > 1)
e) P (X = 1)



−x2 − 3x − 2 ; −2 ≤ x ≤ −1
C sin x ; 0 ≤ x ≤ π2
Example 7 Consider function f (x) = 

0 ; elsewhere
Find
4
a) C
b) F (x)
c) P (− 12 ≤ X ≤ π4 )
d) P (X > 0)
Example 8 Consider function f (x) =
Find
a) C
b) F (x)
c) P (−1 ≤ X ≤ 1)
d) P (X > 2)
5
(
0 ; x≤0
C exp(−2x) ; x > 0
Characteristics of a random variable
Expected value of a random variable X- denoted EX is defined to be
EX =
X
xi pi
i
if X is discrete random variable
Z ∞
EX =
xf (x) dx
−∞
if X is continuous random variable.
Properties
1) If P (X = c) = 1 then EX = c
P (X ≥ 0) = 1 then EX ≥ 0
P (a ≤ X ≤ b) = 1 then a ≤ EX ≤ b
2) If there exists EX then exists E[aX + b], (a, b ∈ R) and
E[aX + b] = aEX + b
3) If there exist EX and EY then exists E[X + Y ] and
E[X + Y ] = EX + EY
4) Let X, Y be the independent random variables. If there exist EX and
EY then exists E[X · Y ] and
E[X · Y ] = EX · EY
5) Let Y = g(X) then
EY =
EY =
X
Z ∞
g(xi )pi
i
g(x)f (x) dx
−∞
The quartile of order p is any number xp such that both P (X ≤ xp ) ≥ p
and P (X ≥ xp ) ≥ 1 − p are true.
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In case of continuous random variable we have F (xp ) = p.
If p = 0.5 then x0.5 is called a median of X.
Variance of random variable X - denoted V arX is defined to be
V arX =
X
i
(xi − EX)2 pi
if X is discrete random variable
V arX =
Z ∞
−∞
(x − EX)2 f (x) dx
if X is continuous random variable.
Properties
1) V arX ≥ 0 (V arX = 0 if P (X = c) = 1)
2) If there exists V arX then exists V ar[aX + b] and
V ar[aX + b] = a2 V arX
3) V arx = EX 2 − (EX)2
4) Let X, Y be the independent random variables. If there exist V arX,
V arY then exists V ar[X + Y ] and
V ar[X + Y ] = V arX + V arY
Standard deviation DX =
√
V arX.
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Some important distributions
1) Binomial distribution
Total number of successes in a series of n independent trials with two
possible outcomes.
!
n k n−k
p q
pk =
k
k = 0, 1, . . . , n
EX = np, V arX = npq
2) Geometric distribution
The number of trials to get the first success, in an independent series of
trials.
pk = pq k−1
k = 1, 2, . . .
EX = p1 , V arX = p1 ( p1 − 1)
3) Poisson distribution
pk =
λk
exp(−λ)
k!
k = 0, 1, . . .
EX = V arX = λ
4) Uniform distribution
f (x) =
EX =
a+b
,
2
V arX =
(
1
b−a
; a<x<b
0 ; elsewhere
(b−a)2
12
5) Exponential distribution
f (x) =
(
1
λ
0 ; x≤0
; x>0
exp(− λ1 x)
EX = λ, V arX = λ2
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6) Normal distribution
1
(x − µ)2
f (x) = √
exp(−
)
2σ 2
2πσ
Notation: X ∼ N (µ, σ)
−∞<x<∞
Properties
a) X ∼ N (µ, σ) ⇒ aX + b ∼ N (aµ + b, a2 σ 2 )
b) Xi ∼ N (µ, σ), are independent then
√
X1 + X2 + . . . + Xn ∼ N (nµ, nσ)
1
σ
(X1 + X2 + . . . + Xn ) ∼ N (µ, √ )
n
n
Theorem
If X ∼ N (µ, σ) then U =
X−µ
σ
∼ N (0, 1)
Example 9 Let X ∼ N (5, 2). Calculate
a) P (|X| < 4)
b) P (|X − 4| ≥ 2)
c) P (3X − X 2 > 0)
7) Chi-square distribution
Let Xi ∼ N (0, 1)
2
χ2m = X12 + X22 + . . . + Xm
8) Student’s or t-distribution
Let X1 ∼ N (0, 1), X2 ∼ χ2m
X1
tm = q
X2
m
9) F-distribution
Let X1 , X2 be independent random variables, such that X1 ∼ χ2k , X2 ∼ χ2m
Fk,m =
9
X1
k
X2
m
Central Limit Theorem
Let {Xk } be a sequence of mutually independent random variables with a
common distribution. Suppose that µ = E[Xk ] and σ 2 = V ar[Xk ] exist and
let Sn = X1 + X2 + . . . + Xn . Then
lim P (
n→∞
Sn − nµ
√
< u) = Φ(u)
σ n
Example 10 We estimate each of 192 numbers. Assume that errorXi of this
estimation for every number has uniform distribution on the interval (-0.5,0.5).
Let S = X1 + X2 + . . . + X192 . Approximate P (|S| < 10).
Example 11 We have 80 bulbs. We use then one by one. Let us assume that
time of living of each this bulbs is a random variable Xi with density function
f (t) =
(
1
1500
0 ; t≤0
1
exp(− 1500
t) ; t > 0
Find probability P (X1 + X2 + . . . + X80 > 100000)
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