Show that E[X|X > K] – Kα/(α-1) for K large enough when a CDF has a tail power of alpha (when 1-F(x) ∝ x-α for x large enough)
Show that E[X|X > K] – Kα/(α-1) for K large enough when a CDF has a tail power of alpha (when ...
3. X is a continuous RV with pdf f(x) and CDF F(x). a) Derive the dist of Y=F(X) b) Show that Z=-21n(Y) has a Gamma dist. & derive it. 4. X-i ~ cont with pdf fi(x) and CDF Fi(x), i=1, 2, , k. all independent. Define YjaFi(Xi), i=1, , k. Derive the distribution of
3. X is a continuous RV with pdf f(x) and CDF F(x). a) Derive the dist of Y=F(X) b) Show that Z=-21n(Y) has a Gamma dist....
d X n oo Show lim sup,P(X>k)0 when X Here cdf of X, and X are continuous for every real number.
d X n oo Show lim sup,P(X>k)0 when X Here cdf of X, and X are continuous for every real number.
A continuous random variable X has cdf F given by: F(x)x3, x e [0,1] (1, x〉1 a) Determine the pdf of X b) Calculate Pi<X <3/4 c) Calculate E X]
2. For a discrete random variable X, with CDF F(X), it is possible to show that P(a < X S b)-F(b) - F(a), for a 3 b. This is a useful fact for finding the probabil- ity that a random variable falls within a certain range. In particular, let X be a random variable with pmf p( 2 tor c-1,2 a. Find the CDF of X b. Find P(X X 5). c. Find P(X> 4). 3. Let X be a...
Let 0 < γ < α . Then a 100(1 − α )% CI for μ when n is large is Xbar+/-zγ*(s/sqrt(n))The choice γ = α /2 yields the usual interval derived in Section 8.2; if γ ≠ α /2, this confidence interval is not symmetric about . The width of the interval is W=s(zγ+ zα-γ)/sqrt(n). Show that w is minimized for the choice γ = α /2, so that the symmetric interval is the shortest. [ Hints : (a)...
3. X is a continuous RV with pdf f(x) and CDF F(x). a) Derive the dist of Y=F(X). b) Show that Z=-2ln(Y) has a Gamma dist. & derive it. 4. X_i ~ cont with pdf f_i(x) and CDF F_i(x), i=1, 2, ..., k. all independent. Define Y_i=F_i(X_i), i=1, ..., k. Derive the distribution of U=-2ln[Y_1.Y_2...Y_k].
Question 6 A random variable X has cdf χ20 Plotthe cdf and identif.,(x)-1-0.2~ a) Plot the cdf and identify the type of the random variable. b) Find the pdf of X. c) Calculate P[-4eX<-1], P(xS2], P(X=1], Pf2-K6], and P[X>10]. d) Calculate the mean and the variance of X. If the random variable X passes through a system with the following chara cteristic function: e) f) Find the pdf of Y. Calculate the mean and the variance of Y. Good Luck
e. A continuous random variable X has cdf $$ F(x)=\left\{\begin{array}{cc} a & x \leq 0 \\ x^{2} & 0< x \leq 1 \\ b & x>1 \end{array}\right. $$a. Determine the constants a and b.b. Find the pdf of X. Be sure to give a formula for fx(X) that is valid for all x. c. Calculate the expected value of X.
6. Consider the problem of solving x/(1 +x)-0.99-0, calling its root a. Then let α (e) be the solution of x/(1 + x)-0.99 + € 0. (a) Using (3.74), estimate (e)- . (b) Calculate α(e)directly, compute α(e)-α, and compare with (a). Comment on your results (3.74 f (a(0)) for all sufficiently small values ofe. If the derivative α' (0) IS very large in size, then the small change e will be magnified greatly in its effect on the root.
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α 5. Let f(x) (3 – x)2 (a) Use ch to show that a power series representation of f is пхп 3n+1 1 - 2 n=0 n=1 00 nxn (b) Find the interval of convergence of 8 WI 3n+1 n=