This document provides answers to the questions in HW0. Feel free to peruse it or use it as a reference.
You should feel very comfortable with everything in this section.
Any given toss of the coin is as likely to be heads as tails.
It is extremely unlikely the coin is fair.
Absolutely not. But I might be willing to bet with someone you don’t know about its flip.
Most prominently, it begs the question, “what is the probability that probability is 1?”. More importantly, it is an utterly unhelpful prediction.
There is some variable X taking values between 0 and 10. 40% of the time it is greater than 6.
The odds are 2 to 3.
E[x] depends on the (mostly unknown) distribution. The max possible value would happen when the distribution is as far right as possible. In this case, that happens when P[x=6]=0.6 and P[x=10]=0.4, giving a max expectation of 7.6. The min is when the distribution is as far left as possible, at P[x=0]=0.6 and P[x=6+epsilon]=0.4, for a minimum expectation of 2.4+epsilon.
E[Y] = 0 says that the mean of Y is 0. E[Y|x=5] = 0 tells us that when x=5, the mean of Y is 0. E[Y|Y=0] is 0.
This is the variance of a binomial, which is np(1-p), or in this case: 25*0.25 = 6.25.
The variance is at a max when the distribution is as spread as possible. That would be when P[x=0]=0.6 and P[x=10]=0.4, for a variance of 24. The min variance is when the distribution is as narrow as possible, when P[x=6]=0.6 and P[x=6+epsilon]=0.4 for a variance of 0.24*epsilon^2. The range of variances possible is (0,24].
The CDF of 2 is 0.6 means that 60% of values (of this unnamed random variable) are 2 or below. The pdf of 2 is 0.6 means that the density (of this unnamed RV) at 2 is 0.6, which is borderline uninterpretable by itself. The two are closely related in that the CDF is the running integral of the density, while the density is the slope of the CDF. Critically, “the pdf of 2 is 0.6” does not mean P[x=2]=0.6. Though “the cdf of 2 is 0.6” does mean P[x<=2] = 0.6.
This section asks for details, but really you should just have some sense of these things, and know how to look up the true answers.
I can’t easily draw this for you, so I’ll plot it in R. You certainly did not need to do that.
Mean = 0. Median = 0. Mode = 0. Variance = 1, SD = 1. Kurtosis = 3 [excess kurtosis = 0]. Skew = 0. 95% predictive interval = (-1.96,1.96) [I would have been happy with (-2,2)].