Assume that the generating function of the nonnegative integer random variable ξ is G(S)
Find the generating functions for the following sequences
(1) an=P{ξ≤n}
(2) bn=P{ξ=2n}
Assume that the generating function of the nonnegative integer random variable ξ is G(S) Find the...
If X is a nonnegative integer-valued random variable then the function P(z), defined for lzl s 1 by is called the probability generating function of X (a) Show that d* (b) With 0 being considered even, show that PX is even) = P(-1) + P(1) (c) If X is binomial with parameters n and p, show that Pix is even) -1+12p (d) If X is Poisson with mean A, show that 1+ e-24 2 P[X is even)- (e) If X...
Q4. Assume that X is a continuous and nonnegative random variable with the cumulative distribution function Fx Find the quantile function of Y-log(X)
Q4. Assume that X is a continuous and nonnegative random variable with the cumulative distribution function Fx Find the quantile function of Y -log(X)
Q. 5. Let X be any random variable, with moment generating function M(S) = E[es], and assume M(s) < o for all s E R. The cumulant generating function of X is defined as A(s) = log Ele**] = log M(s), SER Show the following identities: (1) A'(0) = E[X]. (2) A”(0) = Var(X). (3) A"(0) = E[(X - E[X]))). Using the inversion theorem for MGFs, argue the following: (4) If A'(s) = 0 for all s ER, then P(X=...
9. (9 pts) The random variable r-Gamma(x-2, β-4). functions to prove that the moment generating function for the random variable W mw(t) (1-12t)2. Use the method of moment-generating 3Y +5is eSt 10, (9 pts) Suppose that Y has a gamma distribution with α-n/2 for some positive integer n and β equal to some specified value. Use the method of moment-generating functions to prove that W- 2Y /g has a Chi-squared distribution with n degrees of freedom. Make sure you show...
9. (9 pts) The random variable ??~??????????(∝= 2, ?? = 4). Use
the method of moment-generating functions to prove that the moment
generating function for the random variable ?? = 3?? + 5 is
10.
9. (9 pts) The random variable Y-Gamma(α-2. functions to prove that the moment generating function for the random variable W mw(t)120)2 4). Use the method of moment-generating 3Y 5 is est (1-12t)2 10, (9 pts) Suppose that Y has a gamma distribution with α-n/2 for...
The moment generating function ф(t) of random variable X is defined for all values of t by et*p(x), if X is discrete e f (x)dx, if X is continus (a) Find the moment generating function of a Binomial random variable X with parameters n (the total number of trials) and p (the probability of success). (b) If X and Y are independent Binomial random variables with parameters (n1 p) and (n2, p), respectively, then what is the distribution of X...
5*. Consider all sequences (ai,. .., an) such that a, are nonnegative integers and a ai+ 2. Let P, n and Rn be the number of such sequences which start from 0, 1 and 2 respectively. (a) Compute P, Qn, Rn by writing down all such sequences for n 1,2,3. (b) Prove that P, Qn Rn satisfy the recurrence relations: (c) Translate the above equations into linear equations for the generating functions for P, Qn, Rn (d) Solve these equations...
Problem 3. Let X and Y be two independent random variables taking nonnegative integer values (a) Prove that for any nonnegative integer m 7m k=0 b) Suppose that X~ B (n, p) and Y ~ B(m. p), and X, Y are independent. What is the distribution of the random variable Z X + Y? (c) Prove the following formula for binomial coefficients: n\ _n + m for kmin (m, n) (d) Let X ~ B (n, 1/2). What is P...