2. Let X be a binomial variable with n=10. Suppose E(X) - 3. (a) (5 points) What is the probability of success of the binomial experiment that generates the variable X? Explain your answer. (b) (5 points) Find P(X = 3). (c) (5 points) Find P(3 < X < 7). (d) (5 points) Find P(X 1 )
Let X be a random variable, which has a binomial distribution with parameters n and p. It is known that E(X) = 12 and Var(X) = 4. Find n and p.
If X is a binomial random variable with n and p as indicated, compute the probabilities for each of the following cases: P (X<3), n=8, p=.7
(n) 6. Let X ~ Binomial (n,p). Prove that a. Ex=0 (6)p*(1 – p)n-* = ... = 1 b. E[X] = 21-0 x()p*(1 - 2)^-^ = = mp c. Var[X] = x=0x2 (1)p*(1 – p)n-x – (np)2 = ... = np(1 – p) d. My(t) = ... = (pet + 1 - p)n
(c) Idifficult] Let x ~ Binomial (n, p) where n is an even number. Find the PMF of Y g(X) mod (X, 2) where "mod" denotes modulus division of the first argument by the second argument (d) Idifficult MA] Let X~NegBin (k, p). Find the PMF of Y g(X) = mod(X, n) where n E N
Let X, Y be independent random variables where X is binomial(n = 4, p = 1/3) and Y is binomial(n = 3,p = 1/3). Find the moment-generating functions of the three random variables X, Y and X + Y . (You may look up the first two. The third follows from the first two and the behavior of moment-generating functions.) Now use the moment-generating function of X + Y to find the distribution of X + Y .
If x is a binomial random variable, compute p(x) for each of the cases below. a. n=4, x=1, p=0.4 b. n=6, x=3, q=0.6 c. n=3, x=0, p=0.8 d. n=4, x=2, p=0.7 e. n=6, x=3, q=0.4 f. n=3, x=1, p=0.9
Consider a binomial experiment with n = 8 and P=0.30. a. Compute the probability of two successes P(2). b. Compute the probability of three successes P(3). c. Compute the probability of at least four successes P(x> 4). d. Compute the probability of two or fewer successes P(x < 2). e. Compute the mean E(x). f. Compute the variance and standard deviation Var(x) and 0.
If x is a binomial random variable, compute p(x) for each of the cases below. a. n=5, x=2, p=0.3 b. n=6, x=3, q=0.2 c. n=4, x=1, p=0.7 d. n=5, x=0, p=0.4 e. n=6, x=3, q=0.8 f. n=4, x=2, p=0.6
Let X be a binomial random variable with n = 5 and p = 0.30 Use the Binomial Tables to obtain the correct probability distribution Find each probability. 1) P(X = 5) 2) P(X ?= 1)