{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

MIT6_042JF10_rec22

# MIT6_042JF10_rec22 - j ’s are mutually independent and...

This preview shows pages 1–2. Sign up to view the full content.

6.042/18.062J Mathematics for Computer Science December 3, 2010 Tom Leighton and Marten van Dijk Problems for Recitation 22 1 Properties of Variance In this problem we will study some properties of the variance and the standard deviation of random variables. a. Show that for any random variable R , Var [ R ] = E [ R 2 ] E 2 [ R ]. b. Show that for any random variable R and constants a and b , Var [ aR + b ] = a 2 Var [ R ] . Conclude that the standard deviation of aR + b is a times the standard deviation of R . c. Show that if R 1 and R 2 are independent random variables, then Var [ R 1 + R 2 ] = Var [ R 1 ] + Var [ R 2 ] . d. Give an example of random variables R 1 and R 2 for which Var [ R 1 + R 2 ] = Var [ R 1 ] + Var [ R 2 ] . e. Compute the variance and standard deviation of the Binomial distribution H n,p with parameters n and p . f. Let’s say we have a random variable T such that T = j n =1 T j , where all of the T

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: j ’s are mutually independent and take values in the range [0 , 1]. Prove that Var(T) ≤ Ex(T). We’ll use this result in lecture tomorrow. Hint: Upper bound Var [ T j ] with E [ T j ] using the defnition oF variance in part (a) and the rule For computing the expectation oF a Function oF a random variable. MIT OpenCourseWare http://ocw.mit.edu 6.042J / 18.062J Mathematics for Computer Science Fall 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms ....
View Full Document

{[ snackBarMessage ]}

### Page1 / 2

MIT6_042JF10_rec22 - j ’s are mutually independent and...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online