A congressional committee is composed of 10 members: 6 Republicans and 4 Democrats. On any given vote, members vote independently. Whereas Republicans always vote their party line, Democrats vote their party line with probability .8, and vote Republican with probability .2. On a rainy day only 3 committee members (selected w/o replacement) were present. Let X = number of Democrats present Y = Number who voted Democratic that day Find the marginal pdf of X, i.e., P[X=k] for k=0, 1, 2, 3. Display E[X] and VAR[X].
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
A congressional committee is composed of 10 members: 6 Republicans and 4 Democrats. On any given vote, members vote independently. Whereas Republicans always vote their party line, Democrats vote their party line with probability .8, and vote Republican with probability .2.
On a rainy day only 3 committee members (selected w/o replacement) were present.
Let X = number of Democrats present
Y = Number who voted Democratic that day
- Find the marginal
pdf of X, i.e., P[X=k] for k=0, 1, 2, 3. Display E[X] and VAR[X].
- Determine the conditional probability that P[ Y=j | X=k] for k=3 and j=0,1,2,3
- Display the table of joint probabilities fX,Y(i,j) for 0 ≤ j ≤ i, i=0,1,2,3
- Display the marginal pdf of Y. Is it recognizable? Compute E[Y]and VAR[Y].
- Determine the conditional expectation of Y given that X=x, for x=0,1,2,3.
- Compute the COV(X,Y), possibly using part e.
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