2. X = 2*Bin(300,1/2) – 300 E[X] = 0

3. Y = 2*Bin(30,1/2) – 30 E[Y] = 0

4. Z = 4*Bin(10,1/4) – 10 E[Z] = 0

5. W = 0 E[W] = 0

6. Is there a good parameter that allow to distinguish between these distributions? Is there a way to measure the spread?

7. The variance of X, denoted by Var(X) is the mean squared deviation of X from its expected value  = E(X):  Var(X) = E[(X-  ) 2 ].  The standard deviation of X, denoted by SD(X) is the square root of the variance of X.

8. Proof: E[ (X-  ) 2 ] = E[X 2 – 2  X +  2 ] E[ (X-  ) 2 ] = E[X 2 ] – 2  E[X] +  2 E[ (X-  ) 2 ] = E[X 2 ] – 2  2 +  2 E[ (X-  ) 2 ] = E[X 2 ] – E[X] 2 Claim: Var(X) = E[(X-  ) 2 ].

9. 1.Claim: Var(X) >= 0. Pf: Var(X) =  (x-  ) 2 P(X=x) >= 0 2.Claim: Var(X) = 0 iff P[X=  ] = 1.

12. Theorem: X is random variable on sample space S, and P(X=r) it’s probability distribution. Then for any positive real number r: (proof in book) In words: the probability of finding a value of X farther away from the mean than r is smaller than the variance divided by r^2. r

13. What is the probability that with 100 Bernoulli trials we find more than 89 or less than 11 successes when the prob. of success is ½. X counts number of successes. EX=100 x ½ =50 V(X) = 100 x ½ x ½ = 25. P(|X-50|>=40)<=25/40^2 = 1/64

15.  “k standard deviations”

20.  Letbe random variables that for

22. ,

23.  What is the probability of getting 25 or fewer heads in 100 coin tosses?