Stochastic Processes · 免费预览
期望值与矩
期望值按概率对各个取值加权，方差衡量其周围的离散程度；线性性加上独立性，则使和的均值与方差几乎不费吹灰之力即可算出。
1. Averages weighted by probability
Averages weighted by probability
2. Continuous averages
Continuous averages
3. Linearity and adding variances
Linearity and adding variances
4. Tail sums and generating functions
Tail sums and generating functions
5. The mean of one die
Roll one fair six-sided die. What is the expected value of the outcome, and how is it computed?
6. The variance of one die
Continuing with one fair die, knowing that E X is 3.5 and E X squared is 91 over 6, what is the variance of the outcome?
7. Mean and variance of a geometric count
Repeat independent trials that each succeed with probability one quarter until the first success, and let N count the trials needed. What are the mean and variance of N?
8. Moments of a uniform on zero to two
X is uniform on the interval from 0 to 2. What are its mean and variance?
9. A binomial mean and variance
A multiple-choice quiz has 100 independent questions, each guessed correctly with probability 0.1. For the number of correct answers S, what are the mean and variance?
10. Summing a survival curve
A nonnegative integer-valued X has P(X at least 1) equal to 0.4, P(X at least 2) equal to 0.2, P(X at least 3) equal to 0.05, and P(X at least k) equal to 0 for k at least 4. Use the tail-sum formula to find E X.
11. The mean of a die
Roll one fair six-sided die and compute the expected value of the face shown. Write the number as a decimal or a fraction.
12. Variance of a uniform interval
X is uniform on the interval from 0 to 2. Compute the variance of X and write the exact fraction.
13. Mean of a binomial count
A fair coin is flipped 16 times and S counts the heads. Compute the expected number of heads and write the number.