3. Let X be a geometric random variable with parameter p. Prove that P(X >k+r|X >...
Let X be an exponentially distributed random variable with parameter λ. Prove that P(X > s + tK > t) P(X > s) for any S,12 0
Problem 8 (10 points). Let X be the random variable with the geometric distribution with parameter 0 <p <1. (1) For any integer n > 0, find P(X >n). (2) Show that for any integers m > 0 and n > 0, P(X n + m X > m) = P(X>n) (This is called memoryless property since this conditional probability does not depend on m. Dobs inta T obabilita ndomlu abonn liaht bulb indofootin W
A discrete random variable X follows the geometric distribution with parameter p, written X ∼ Geom(p), if its distribution function is A discrete random variable X follows the geometric distribution with parameter p, written X Geom(p), if its distribution function is 1x(z) = p(1-P)"-1, ze(1, 2, 3, ). The Geometric distribution is used to model the number of flips needed before a coin with probability p of showing Heads actually shows Heads. a) Show that fx(x) is indeed a probability...
Let X be the random variable with the geometric distribution with parameter 0 < p < 1. (1) For any integer n ≥ 0, find P(X > n). (2) Show that for any integers m ≥ 0 and n ≥ 0, P(X > n + m|X > m) = P(X > n) (This is called memoryless property since this conditional probability does not depend on m.)
Let X be a geometric random variable with p = 0.83. Use your calculator to find PX < 4)
Let X be a discrete random variable with values in N = {1, 2,...}. Prove that X is geometric with parameter p = P(X = 1) if and only if the memoryless property P(X = n + m | X > n) = P(X = m) holds. To show that the memoryless property implies that X is geometric, you need to prove that the p.m.f. of X has to be P(X = k) = p(1 - p)^(k-1). For this, use...
3. Let X be a random variable and denote by Mx(t) its MGF. Prove that, for any t > 0, we have P[X >Mx(t)e
2. Let X be an exponentially distributed random variable with parameter 1 = 2. Determine P(X > 4). 3. Let X be a continuous random variable that only takes on values in the interval [0, 1]. The cumulative distribution function of X is given by: F(x) = 2x² – x4 for 0 sxsl. (1) (a) How do we know F(x) is a valid cumulative distribution function? (b) Use F(x) to compute P(i sX så)? (c) What is the probability density...
9. Let X be a Poisson random variable with parameter k = 3. (a) P[X 25] (b) Find P[5 S X <10) (c) Find the variance ? 10. Use the related Table to find the following: (here Z represents the standard normal variable) (a) P[Z > 2.57] (b) The point z such that PL-2 SZ sz]=0.8
2. (D5) Let n = o(a) and assume that a =bk. Prove that <a >=<b> if and only if n and k are relatively prime.