If Y is a subspace of a vector space X and f is a linear functional on X such that f(Y) is not the whole scalar field of X. Show that f(y)=0 for all y∈Y.

Suppose that a random sample is to be taken from a normal distribution for which the value of the mean θ is unknown and the standard deviation is 2, and the prior distribution of θ is a normal distribution for which the standard deviation is 1. What is the smallest number of observations that must be included in the sample in order to reduce the standard deviation of the posterior distribution of θ to the value 0.1?

Suppose that there is a probability of 1/50 that you will win a certain game. If you play the game 50 times, independently, what is the probability that you will win at least once?

Suppose that A, B, and D are events such that A and B are independent, Pr(A ∩ B ∩ D) = 0.04, Pr(D|A ∩ B) = 0.25, and Pr(B) = 4 Pr(A). Evaluate Pr(A ∪ B).

Suppose that a random variableX has the binomial distribution with parameters n = 8 and p = 0.7. Find Pr(X ≥ 5)

Suppose that a random variable X has the binomial distribution with parameters n = 15 and p = 0.5. Find Pr(X < 6).

A civil engineer is studying a left-turn lane that is long enough to hold seven cars. Let X be the number of cars in the lane at the end of a randomly chosen red light. The engineer believes that the probability that X = x is proportional to (x + 1)(8 − x) for x = 0, . . . , 7 (the possible values of X).
a. Find the p.f. of X.
b. Find the probability that X will be at least 5.

Suppose that two balanced dice are rolled, and let X denote the absolute value of the difference between the two numbers that appear. Determine and sketch the p.f. of X.

Let f0(x) be the p.f. of the Bernoulli distribution with parameter 0.3, and let f1(x) be the p.f. of the Bernoulli distribution with parameter 0.6. Suppose that a single observation X is taken from a distribution for which the p.d.f. f (x) is either f0(x) or f1(x), and the following simple hypotheses
are to be tested:
H0: f (x) = f0(x),
H1: f (x) = f1(x).
Find the test procedure δ for which the value of α(δ)+β(δ) is a minimum.

Show that the norm ||x|| of x is the distance from x to 0

Show that the set X of all integers with metric d defined by d(m,n)=|m-n| is a complete metric space.

Let a,b ∈ R and a < b. Show that the open interval (a,b) is an incomplete subspace of R, whereas the closed interval [a,b] is complete.

If a man has six different sportshirts and four different  pairs of slacks, how many different combinations can he wear?

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A man has 8 pairs of pants, 12 shirts, 15 ties, and 6 sport coats. If he wears one of each, how
many different outfits can he wear?

In how many ways different ways can the five letters a,b,c,d, and e be arranged?

Answer is: 5! = 120 ways

In how many ways different ways can the five letters a,b,c,d, and e be arranged?

Answer is: 5! = 120 ways

Test the standard normal pseudo-random number generator on your computer by generating a sample of size 10,000 and drawing a normal quantile plot. How straight does the plot appear to be?