6. Let F be the CDF of a random variable X. Prove that lim F(t) =...
(a) Let X be a continuous random variable with the cdf F(x) and pdf f(.1). Find the cdf and pdf of |X|. (b) Let Z ~ N(0,1), find the cdf and pdf of |Z| (express the cdf using ” (-), the cdf of Z; give the explicit formula for the pdf).
Proble 2. Let Fx(t) be the cumulative distribution function (CDF) of a continuous random variable X and let Y-X. Express the CDF of Y terms of Fx(t).
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 f(x)={user user = { x 8. Prove the following 10 a. Prove lim f(x) = 0 b. Prove lim f(x)=1 c. Prove lim f(x) does not exist. 1-2
A random variable X is normally distributed. Let F (x) be the CDF of X. Observations of a very large sample size shows that F (20.21) = 0.025 and F(41.63) = 0.975. Determine the following probability: P (X < 35.00). Hint: for a normal distribution, about 95% of the scores falls within plus or minus two standard deviations from the mean.
Let X be a continuous random variable with CDF Fx and expected value E[X] = 4. Show that (1 Fx(t))dt Fx(t)dt 0 Remark: Make sure to justify - for example with a picture - any manipulations for multiple integrals Let X be a continuous random variable with CDF Fx and expected value E[X] = 4. Show that (1 Fx(t))dt Fx(t)dt 0 Remark: Make sure to justify - for example with a picture - any manipulations for multiple integrals
Let X be a random variable with support Sx = [−6, 3] and pdf f(x) = 1/81x^2 for x ∈ SX , 0 otherwise. Consider the random variable Y = max(X, 0). Calculate the CDF of Y , FY (y), where y is any real number.
1. Let X be a continuous random variable with CDF F(ro)-a+b 3 and support set 0, 1]. (a) Calculate the values of a, b that would make F(ro) a valid CDF. (b) Write out the pdf of X. c) Calculate EX d) Calculate EX
1. Suppose that X is continuous random variable with PDF f(x) and CDF F(x). (a) Prove that if f(x) > 0 only on a single (possible infinite) interval of the real numbers then F(x) is a strictly increasing function of x over that interval. [Hint: Try proof by contradiction]. (b) Under the conditions described in part (a), find and identify the distribution of Y = F(x). 2. Suppose now that X ~ Uniform(0, 1). For each of the distributions listed...
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