Problem 4. Let X be normally distributed with mean 1 and variance 2.2. (a) Find P0.5 < X < 2). (b) Find 95th percentile of X.
1) Suppose X is a Normal RV with mean = 12 and variance = 16. Find (a) P(X < 14) (b) P(14.5 < X < 18) (c) P(X < 16 or X > 12). Hint: Remember to always identify outcomes of interest first! (d) The center of the probability density function of X.
1. Using calculus, find the mean and variance of a uniform distribution with a minimum value of of O and a maximum value of 10. (Give a proof.) Remember that the variance can be calculated using: < X z >-< X >2.
(10pts) 2. Assume X is normally distributed with a mean of 5 and variance of 16. Determine the value of x that solves each of the following: P(x < X < 9) = 0.2 b) P(-x < X-5<x) = 0.99
2) Suppose X is a Normal RV with mean = 17 and variance = 4. Find (a) P(X < 14) (b) P(14.5 < X < 18) (c) P(X < 11 or X > 17) (d) P(X < 11 and X > 17)
Prove that A = B for: A = {(x,y) e Rº : +y/<1} B = {(z,y) € RP: (71+ y)² < 1}
3(8r - , 0<x<4 Determine the mean and variance of the random variable for f(x) Round your answers to two decimal places (e.g. 98.76) E(X) VOX) = Click if you would like to Show Work for this question: Open Show Work 128
The random variable Z has a Normal distribution with mean 0 and variance 1. Show that the expectation of Z given that a < Z < b is o(a) – °(6) 0(b) – (a)' where Ø denotes the cumulative distribution function for Z.
5. Find the Fourier Transform of g(t) = {o. (1-x?, x<1, 1</z/.
1. Let X1, ..., Xn be random sample from a distribution with mean y and variance o2 < 0. Prove that E[S] So, where S denotes sample standard deviation. 10 points