Stochastic Processes · 免费预览
随机变量与分布
随机变量把结果转化为数值，而它们的分布——离散情形下的二项分布、几何分布和泊松分布，连续情形下的均匀分布、指数分布和正态分布——构成了后续每一章都要用来计算的词汇。
1. From outcomes to numbers
From outcomes to numbers
2. Densities for continuous values
Densities for continuous values
3. The distribution function
The distribution function
4. Joints, marginals and sums
Joints, marginals and sums
5. A binomial probability
A fair coin is flipped three times and S counts heads. Using the binomial distribution with n equal to 3 and p equal to one half, what is the probability of exactly two heads?
6. A geometric probability
A fair coin is flipped until the first head appears. What is the probability that the first head comes on exactly the third flip?
7. Poisson at zero
Let X have a Poisson distribution with parameter lambda equal to 2. Which expression gives the probability that X equals 0?
8. Exponential cumulative probability
Let X be exponential with rate lambda equal to 2. Which expression gives the probability that X is at most 1?
9. Interval probability from the distribution function
X is uniform on the interval from 0 to 4. Using the distribution function, what is the probability that X lies between 1 and 3?
10. The density of a sum at a point
U and V are independent and uniform on the interval from 0 to 1. Using the triangular density of the sum, what is the density of U plus V evaluated at 1.5?
11. Exactly one head in three flips
A fair coin is flipped three times. Compute the probability of exactly one head using the binomial distribution and write the number, either as a decimal or as a fraction.
12. A uniform tail probability
X is uniform on the interval from 0 to 4. Compute the probability that X is at most 1 and write the number, either as a decimal or as a fraction.
13. Height of the triangular density
U and V are independent uniforms on the interval from 0 to 1. Compute the density of U plus V at the point 0.5 and write the number, either as a decimal or as a fraction.