5. Let W Geo() (a) Determine the probability that W is even. (b) Determine the probability...
Problem 6, 15 Points. Let ~ ?(A), the exponential distribution. Determine and identify the probability distribution of the random variable [x], where fz] denotes the ceiling function, the smallest integer greater than or equal to z. Remark. Note that the new random variable is discrete random variable.
Let n be an even positive integer. What is the probability that a randomly chosen n-bit string has the same number of zeros and ones? (Please show work)
a. Let A = { < A,w > | A is a DFA that accepts w}, M is a Turing machine, and L(M) = A. Suppose M accepts the string p. p is in the form of < B,s > where B is a DFA, s is a string, and B accepts s. True False b. A linear equation is in the form of ax + b where a and b are constants and x is a variable. Let x-intercept...
5. (a) Show that 26 = 1 mod 9. (b) Let m be a positive integer, and let m = 6q+r where q and r are integers with 0 <r < 6. Use (a) and rules of exponents to show that 2" = 2 mod 9 (c) Use (b) to find an s in {0,1,...,8} with 21024 = s mod 9.
are even, r even. A 1.1-14·Let the interval [-r,r] be the base of a semicircle. ne of the f a pot is selected at random from this interval, assign abilty of a probability to the event that the length of the perpen- e slot into dicular segment from the point to the semicircle is less than r/2. 1.1-15. Let S = A1 U A2 U U Am, where events (a) If P(A1) P(A2)P(A), show that P(Ai - (b) If A...
A single die is rolled twice. Find the probability of rolling an even number the first time and a number greater than 4 the second time. Find the probability of roing an even number the first time and a number greater than 4 the second time. (Type an integer or a simplified fraction.)
Let L = { w∈ {a, b}∗|w has even length and contains the substring aba } Design NFA.
can you compute? It is not necessary to actually compute each probability. 5. Let Q11,2,3,... be the countably infinite sample space whose elements (outcomes) are the positive integers. For each positive integer n, define the event An = { k : k is a multiple of n } a) Findn and m such that An- A3n A4 and Am A6n A9 b) Pk) (3)find the probabilty of the event As 0.210526.. you will receive no credit.
0.5 0 0 5. Let P 0.5 0.6 0.3represent the probability transition matrix of a Markov chain with three 0 0.4 0.7 states (a) Show that the characteristic polynomial of P is given by P-ÀI -X-1.8λ2 +0.95λ-0.15) (b) Verify that λι 1, λ2 = 0.5 and λ3 = 0.3 satisfy the characteristic equation P-λ1-0 (and hence they are the eigenvalues of P) c) Show thatu3u2and u3are three eigenvectors corresponding to the eigenvalues λι, λ2 and λ3, respectively 1/3 (d) Let...
8. Let w cos(2π/5) + isin(2π/5). Here we describe how to express w in terms of square roots. (a) Show that w is a root of the polynomial 24+2+22+21. Hint: 25-1-(-(24+23+22+2+1) (b) Show that w + is a root of the polynomial u2 + u-1 (c) Show that Ve, where V5 means the positive square root of Hint: Figure out the sign of w by adding the polar forms of w and 1/w. (d) Put β--12vS So in part (c),...