4. (20 pts.) [Bonus Question) Suppose that X is a Binomial RV with p=0.5 i.e. X...
5. (20 pts) Function of RV Let Ry X-Exponential(1),i.e.,the CDF is Fx (x) = (1 - )u(x). IEX = 9(x) = -2x + 1, find the CDF Fy (y) and the PDF fy(y).
7 Suppose X~bin(n 20, p = .4). Derive the closed form (i.e. no long sum) for Mr(t) We were unable to transcribe this image Suppose X~bin(n 20, p = .4). Derive the closed form (i.e. no long sum) for Mr(t)
Suppose X and Y are independent Binomial random variables, each with n=3 and p=9/10. a. Find the probability that X and Y are equal, i.e., find P(X=Y). b. Find the probability that X is strictly larger than Y, i.e., find P(X>Y). c. Find the probability that Y is strictly larger than X, i.e., find P(Y>X).
binomial RV B(n,p) 2. Simulating a Binomial RV. One procedure for generating uses n EXi is binomial if realizations of a uniform random variable and exploits the fact that Y the Xi are Bernoulli RVs. Here is an alternative procedure that requires generating only a single (!) uniform variate: 1/p and B 1/(1 p) 0) Let 1) Set 0 U[0, 1] 2) Generate 3) If k n, go to step 5; else, k ++ au; if u B(u- p). Go...
Problem 5. Let X be a binomial random variable with parameters n and p. Suppose that we want to generate a random variable Y whose probability mass function is the same as the conditional mass function of X given X-k, for some k-n. Let a = P(X-k), and suppose that the value of a has been computed (a) Give the inverse transform method for generating Y. (b) Give a second method for generating Y (c) For what values of a,...
Question # A.4 (a) Given that probability density function (pdf of a random variable (RV), x is as follows: Px(x)-axexp(-ax) x 20 otherwise where α is a constant. Suppose y = log(x) and y is monotonic in the given range of X. Determine: (i) pdf of y; (ii) valid range of y; and, (iii) expected value of y. Answer hint:J exp(y) (b) Given that, the pdf, namely, fx(x) of a RV, x is uniformly distributed in the range (-t/2, +...
Let X ∼ Bin(n = 20, p = 0.25) be a Binomial r.v. with parameters n = 20, p = 0.25. Given that X does not exceed 5, what is the probability that X takes an even value?
1) Binomial distribution, f(x) = px (1 – p) n-x , x = 0, 1, 2, …, n n = 10, p = 0.5, find Probabilities a) P(X ≥ 2) b) P(X ≤ 9) 2) f(x) = (2x + 1)/25, x = 0, 1, 2, 3, 4 a) P(X = 4) b) P(X ≥ 2) c) P(X ≥ -3) 3) Z has std normal distribution, find z a) P(-1.24 < Z < z) = 0.8 b) P(-z < Z <...
4) (20 pts) Let X be a RV with the following PDF: fx(x) = že=fal for all x. Let Y = X?. (a) Compute E[X]. (b) Find the PDF of Y, fy(y). (c) Compute E[Y].
Show that the binomial probability function is unimodal in x (i.e., p(x) either always increases in x; or only decreases in x; or increases to a maximum and then decreases as x goes from 0 to n). Hint: Consider p(x)/p(x-1).