Question

In: Statistics and Probability

For each positive integer, n, let P({n}) =(1/2^n) . Consider the events A = {n :...

For each positive integer, n, let P({n}) =(1/2^n)
. Consider the events
A = {n : 1 ≤ n ≤ 10}, B = {n : 1 ≤ n ≤ 20}, and C = {n : 11 ≤ n ≤ 20}. Find (a) P(A),
(b) P(B), (c) P(A ∪ B), (d) P(A ∩ B), (e) P(C), and (f) P(B′).

Hint: Use the formula for the sum of a geometric series

Solutions

Expert Solution


Related Solutions

Let n be a positive integer. Let S(n) = n sigma j=1 ((1/3j − 2) −...
Let n be a positive integer. Let S(n) = n sigma j=1 ((1/3j − 2) − (1/3j + 1)). a) Compute the value of S(1), S(2), S(3), and S(4). b) Make a conjecture that gives a closed form (i.e., not a summation) formula for the value of S(n). c) Use induction to prove your conjecture is correct.
Let A[1..n] be an array of distinct positive integers, and let t be a positive integer....
Let A[1..n] be an array of distinct positive integers, and let t be a positive integer. (a) Assuming that A is sorted, show that in O(n) time it can be decided if A contains two distinct elements x and y such that x + y = t. (b) Use part (a) to show that the following problem, re- ferred to as the 3-Sum problem, can be solved in O(n2) time: 3-Sum Given an array A[1..n] of distinct positive integers, and...
Let n be a positive integer. Prove that if n is composite, then n has a...
Let n be a positive integer. Prove that if n is composite, then n has a prime factor less than or equal to sqrt(n) . (Hint: first show that n has a factor less than or equal to sqrt(n) )
6. Let t be a positive integer. Show that 1? + 2? + ⋯ + (?...
6. Let t be a positive integer. Show that 1? + 2? + ⋯ + (? − 1)? + ?? is ?(??+1). 7. Arrange the functions ?10, 10?, ? log ? , (log ?)3, ?5 + ?3 + ?2, and ?! in a list so that each function is big-O of the next function. 8. Give a big-O estimate for the function ?(?)=(?3 +?2log?)(log?+1)+(5log?+10)(?3 +1). For the function g in your estimate f(n) is O(g(n)), use a simple function g...
please provide good answer. 1). Let S = { 1,2,3, …., n} for some positive integer...
please provide good answer. 1). Let S = { 1,2,3, …., n} for some positive integer n. Define the operations + and . on S as x + y = max{ x, y }, and x.y = min{ x, y }. Is it possible to make S into a Boolean algebra with these two operations? Explain your reasoning. [ Note: max{ x, y } returns the maximum of the values x and y, and min{ x, y } returns the...
Prove that τ(n) < 2 n for any positive integer n. This is a question in...
Prove that τ(n) < 2 n for any positive integer n. This is a question in Number theory
Prove the following theorem. If n is a positive integer such that n ≡ 2 (mod...
Prove the following theorem. If n is a positive integer such that n ≡ 2 (mod 4) or n ≡ 3 (mod 4), then n is not a perfect square.
A Mystery Algorithm Input: An integer n ≥ 1 Output: ?? Find P such that 2^p...
A Mystery Algorithm Input: An integer n ≥ 1 Output: ?? Find P such that 2^p is the largest power of two less than or equal to n. Create a 1-dimensional table with P +1 columns. The leftmost entry is the Pth column and the rightmost entry is the 0th column. Repeat until P < 0 If 2^p≤n then put 1 into column P set n := n - 2^p Else put 0 into column P End if Subtract 1...
Let G be an abelian group and n a fixed positive integer. Prove that the following...
Let G be an abelian group and n a fixed positive integer. Prove that the following sets are subgroups of G. (a) P(G, n) = {gn | g ∈ G}. (b) T(G, n) = {g ∈ G | gn = 1}. (c) Compute P(G, 2) and T(G, 2) if G = C8 × C2. (d) Prove that T(G, 2) is not a subgroup of G = Dn for n ≥ 3 (i.e the statement above is false when G is...
7. Let n ∈ N with n > 1 and let P be the set of...
7. Let n ∈ N with n > 1 and let P be the set of polynomials with coefficients in R. (a) We define a relation, T, on P as follows: Let f, g ∈ P. Then we say f T g if f −g = c for some c ∈ R. Show that T is an equivalence relation on P. (b) Let R be the set of equivalence classes of P and let F : R → P be...
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT