Suppose that a random variable X :S → R is such that the set X(S) is countably infinite, that is, X(S) = {xk : k E N}. Then the sum EP(X = ®x) k=0 is equal to Answer:
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- Prove that the maximal entropy of a discrete random variable is logan (n being the number of possible values of the random variable) and is attained for P₁ = P2 = = P₁ = 1/n.Let X be a continuous random variable described below: p(X) = x/4, for all x in the range 0 ≤ x ≤ 2 = x2/8, for all x in the range 2 < x ≤ k (a) What is the value of k?(b) What is E(X)? Note: Answers to both questions are irrational numbers.Show that the CDF of a geometric random variable with parameter p can be expressed as
- 2.6. Let (2, F, P) be a probability space and let A1, A2,... be sets in F such that P(Ag) < ∞ . k=1 Prove the Borel-Cantelli lemma: P(N U Ak) = 0, m=1 k=m i.e. the probability that w belongs to infinitely many A{s is zero.Consider a Poisson process of intensity λ > 0 events per hour. Suppose that exactly one event occurs in the first hour. Let X be the time at which this event occurs, measured in hours (so that the possible values of X are the reals in the interval [0, 1]). (a) What is P (0 ≤ X ≤ a) for a ∈ [0, 1]? (Of course, we are conditioning on exactly one event occurring in the first hour, as described.) Hint: Consider the number of events that occur in [0, a] and in (a, 1], similar to question #5 on the previous homework. (b) What is the distribution of X, under the same conditions? (You might want to wait until Monday’s class after break to do this last part.)b) A continuous random variable X has the p.d.f f(x) = {A(2 – x)(2 + x), 0 < x < 2, Find (i) the value of A, (ii)P(X <1)(iii) P(1 < X <2). l0, otherwis ------------
- 4) Let X and X1, X2, ... be random variables. Then - Xn 4 X as n → ∞ + E → 0 as n → 0. 1+|Х, — |x - "x|3. Let the random variable X have the pdf f(x) = 2(1 — x), 0 ≤ x ≤ 1, zero elsewhere. a) Find the cdf of X. Provide F(x) for all real numbers x (set up the appropriate cases). b) Find P(1/4 < X < 3/4). c) Find P(X= 3/4). d) Find P(X ≥ 3/4).Let pX(x) be the pmf of a random variable X. Find the cdf F(x) of X and sketch its graph along with that of pX(x) if pX(x)=1/3,x=−1,0,1, zero elsewhere
