Random Variables and Probability : Sampling Without Replacement
A carton of 30 lightbulbs includes 5 defective ones. If 4 light bulbs are drawn at random (without replacement), what is the probability that; (a) 2 of the selected light bulbs are defective. (b) Not all the selected light bulbs are defective.
Probability : Drawing Cards and Sampling Without Replacement
3 cards are drawn in succession from a regular straight deck of 52 playing cards. Find the probability that: (a) the first card is a Red Ace. (b) the second card is a 10 or Jack. (c) the third card is greater than 3 but less than 7.
We are using the book Methods of Real Analysis by Richard R. Goldberg (See attached file for full problem description) --- 12.5-2 Show that the Fourier series for is a) Use 12.5E to show that Fourier series at t=0 converges to . Deduce that 12.5E: Theorem. Let ( this ...continues
We are using the book Methods of Real Analysis by Richard R. Goldberg (See attached file for full problem description) --- 12.6-3 Let be a complete orthogonal family in . Define the function A from into .( This means: In order to manufacture our metric space we must therefore regard any two function whose valu ...continues
(See attached file for full problem description) --- 12.6-1 Calculate the Legendre functions and show that they are orthogonal to one another on [-1,1] and that each has norm equal to 1.
Let (X,B,mu) be a complete, finite measuable space. For each C,D in B, set d(C,D) = mu (C / D) where C / D is the symmetric difference of C and D. We say that two measurable sets C,D are equivalent if d(C,D)=0 (this is an equivalence relation). Let E be the set of equivalence classes, and show that d introduces a metri ...continues
Almost every point is a density point
A point x of a measurable subset A of the reals is called a density point if m( A intersection [x-h, x+h] ) / 2h goes to 1 as h goes to 0 where m is the Lebesgue measure. Prove that if A is a set of positive, finite Lebesgue measure, then almost every point of A is a density point. I would like to note that I can use ...continues
Properties of additive functions; Bounded; Continuous; Measurable
Let f : R --> R be an additive function i.e. f(x+y) = f(x) + f(y) for all x,y in R. 1. If f is bounded at a point, then f is continuous at that point. 2. If f is measurable, then f is linear i.e. f(x) = cx for some c in R. I have already proved that f is continuous if and only if f is linear, and I have proven that if f i ...continues
Space of functions is sequentially compact
Let C_0 be the space of functions f:R --> R such that lim f(x) = 0 as x goes to infinity and negative infinity C_0 becomes a metric space with sup-norm ||f|| = sup { |f(x)| : x in R } Prove that if A is a family of functions in C_0 such that A is uniformly bounded and equicontinuous, then every sequence of functions ...continues
Lebesque measurable sets in R^n.
Prove that lebesque measurable sets in R^n form sigma algebra. ( Please use basic definition when you talk about the lebesgue measurable sets in R^n). The def we have is: (k_1)^(m)={ -1/2 + m_i =< x_i =< 1/2+ m} m=(m_1,m_2,...,m_n) m belongs to z^d Now we say that A in R^n is Lebesque measurable set in R^n if ...continues
