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SupposeYi,Yexp(e Use the CLT to approximate the following probability P(-1.96 < 1.96)
2. Approximate the following probabilities for sufficiently large n by applying CLT. Please provide answers using the standard CDF Φ(z) = P(Z 2) where Z ~ N(0,1), instead of real numbers. (a) For X Binomial(n, 1/4), P(X/Vn 0.5). (b) For X1, , , , Xn iid Uniform(0, 1), PK, < 2/(3 ) (c) For Y ~ χ2(n), P(Y < n).
3. P(z<zc)=0.95. Find ze (a) 1.28 (b) 1.645 (c) 1.96 (d) -1.645
4, (10 points) Use the reflection principle to find the probability P(Ma = 6), where MS- maxo<iG8S, and ( Snhoo is a simple symmetric random walk starting in 0 (S0 = 0).
Let X be an exponential random variable such that P(X<26) = P(X > 26). Calculate E[X|X > 28]. Answer: CHECK
Data analysis
5. (10 points) Please determine the following probability given the Z value using the standard normal distribution table a) P(Z < 1.28) b) P(Z>1.45)
1.1.11 Express the confidence interval (0.054,0.152) in the form of p-E<p<P+E. [<p< (Type integers or decimals.)
5. Stirling's approximation gives an approximate value for the factorial n! n! xVannne-n n! V2πηη"e-n which means that lim Moreover. V2rnnne--+ 1/(12叶1) < n! < V2mn"e-n+1/12n for all n>1 . a- Show that lim b- We toss a fair coin 100 times. Find an approximate value for the following probability P(we get exactly 50 heads ).
4.28 If Z ~ N(0,1), find the following probabilities: a. P(Z <1.38) b. P(Z > 2.14) c. P(-1.27 <Z<-0.48)
Express the confidence interval 0.666<p<0.888 in the form DIE +E=0+0
The random variable X has the probability density function (x)a +br20 otherwise If E(X) 0.6, find (a) P(X <름) (b) Var(x)