Download Discrete Probability Variance of a random variable R. Inkulu http

Discrete Probability
Variance of a random variable
R. Inkulu
http://www.iitg.ac.in/rinkulu/
(Variance of a random variable)
1/9
Moment of a random variable
Let X be a random variable with probability distribution {f (xj )}, and let r ≥ 0
be P
an integer. The rth moment of X (about the origin) is E(X r ), which is equal
to j xjr f (xj ).1
1
for our purposes, it suffice to assume that E(X r ) exists
(Variance of a random variable)
2/9
Variance and standard deviation
Let E(X) and E(X 2 ) be the first and second moments of a random variable X.
Then the variance (a.k.a. dispersion) of X, denoed with Var(X) or σX2 , is
defined as E((X − E(X))2 ).
- characterizes how widely a random variable is distributed:
small variance indicates large deviations of X from µ are improbable
large variance indicates that not all values assumed by X lie near the mean
The standard deviation of X, denoted by σX , is
p
Var(X)
- measures how spread out the distribution of X around its mean; useful as
its units are same as E(X)
(Variance of a random variable)
3/9
Few properties of variance
• Var(X) = E(X 2 ) − (E(X))2
• Var(aX + b) = a2 Var(X)
• If X has mean µ and variance σ 2 , then X − µ has mean 0 and variance
σ 2 , and hence the variable X ∗ = X−µ
σ has mean 0 and variance 1. (The
∗
variable X is called the normalized variable corresponding to X.)
(Variance of a random variable)
4/9
Examples
• If X assumes the values ±c, each with probability 21 , then Var(X) is
(Variance of a random variable)
5/9
Examples
• If X assumes the values ±c, each with probability 21 , then Var(X) is c2 .
(Variance of a random variable)
5/9
Examples
• If X assumes the values ±c, each with probability 21 , then Var(X) is c2 .
• If X is the number of points scored with a symmetric die, then Var(X) is
(Variance of a random variable)
5/9
Examples
• If X assumes the values ±c, each with probability 21 , then Var(X) is c2 .
• If X is the number of points scored with a symmetric die, then Var(X) is
1 2
6 (1
+ 22 + . . . + 62 ) − ( 27 )2 .
(Variance of a random variable)
5/9
Covariance: definition
The covariance of two random variables X and Y, denoted with Cov(X, Y), is
E[(X − E[X])(Y − E[Y]).
(Variance of a random variable)
6/9
Few properties
• Cov(X, Y) = E(X, Y) − E(X)E(Y)
If X and Y are independent, Cov(X, Y) = 0.
• Let X1 , X2 , . . . , Xn be random variables.
Var(X1 + X2 + . . . + Xn ) =
Pn
k=1 Var(Xk )
+2
P
j<k
Cov(Xj , Xk ).
• Bienayme’s formula: If the Xj are mutually independent, then
Var(X1 + X2 + . . . + Xn ) = Var(X1 ) + Var(X2 ) + . . . + Var(Xn ).
(Variance of a random variable)
7/9
Example
(Variance of a random variable)
8/9
Example
(Variance of a random variable)
9/9