A continuous random variable X has cdf F given by: F(x)x3, x e [0,1] (1, x〉1...
3. (10 points) Let X be a continuous random variable with CDF for x < -1 Fx(x) = { } (x3 +1) for -1<x<1 for x > 1 and let Y = X5 a. (4 points) Find the CDF of Y. b. (3 points) Find the PDF of Y. c. (3 points) Find E[Y]
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Suppose that the continuous random variable X has pdf given by: x <1 0.16x 15 x 33 f(x)= 0.06 3<x55 [124 x>5 • Find the corresponding cdf for X: You must determine the arbitrary constants. x <1 1<x3 Ex(x)={ 3< x <5 x>5 • Use the cdf to find P(2.4 <x< 10) = • Use the cdf to determine the following percentiles: the 50th percentile (median) the 80th percentile the 90th percentile
2. Suppose X is a continuous random variable with the probability density function (i.e., pdf) given by f(x) - 3x2; 0< x < 1, - 0; otherwise Find the cumulative distribution function (i.e., cdf) of Y = X3 first and then use it to find the pdf of Y, E(Y) and V(Y)
2. A random variable X has a cdf given by F(x) = 1 . x < 0 0 < x < 1 <3 x > 3 11, (f) What is P(X = 1)? (g) Find E(X), the expectation of X. (h) Find the 75th percentile of the distribution. Namely, find the value of 70.75 SO that P(X < 70.75) = F(710.75) = 0.75. (i) Find the conditional probability P(X > X|X > 3).
9.) Suppose that X is a continuous random variable with density C(1- if [0,1] px(x) ¡f x < 0 or x > 1. (a) Find C so that px is a probability density function. (b) Find the cumulative distribution of X (c) Calculate the probability that X є (0.1,0.9). (d) Calculate the mean and the variance of X
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
5. (Discrete and ontinuous random variables) (a) Consider a CDF of a random variable X, 10 x < 0; Fx(x) = { 0.5 0<x< 1; (1 x > 1. Is X a discrete random variable or continuous random variable? (b) Consider a CDF of a random variable Y, 1 < 0; Fy(y) = { ax + b 0 < x < 1; 11 x >1, for some constant a and b. If Y is a continuous random variable, then what...
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Additional Problem 3. If X is a continuous random variable having cdf F, then its median is defined as that value of m for which F(m) 0.5. Find the median for random variables with the following density functions (a) f(x) = e-*, x 0 (b) f(x) = 1, 0 〈 x 1. (c) f(x) 6x(1 - x),0 <1.
Additional Problem 3. If X is a continuous random variable having cdf F, then its median is defined as that value of m for which F(m) = 0.5. Find the median for random variables with the following density functions (a) f(r)-e*, x > 0 (c) f(x) 6r(1-x), 1. Additional Problem 6. Let X be a continuous random variable with pdf (a) Compute E(X), the mean of X (b) Compute Var(X), the variance of X. (c) Find an expression for Fx(r),...
E. Consider a continuous random variable X with cdf F(x) = x3/8, 0 < x < 2. (27) The pdf f(x) of X is (а) 6х (b) x3/8 (c) 3x2/8 (d) x2/4(28) E[X2+3X] is (а) 6.9 (b) 4.3 (с) 4.5 (d) 8.1 (29) The probability P(X > 1) is (a) 7/8 (b) 4/8 (c) 6/8 (d) 3/8