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3-) Let ocr<1 o w UUUUU is probability destiny function of X random variable. a- )...
4. Let X be a continuous random variable with probability density function: x<1 0, if if| if x>4 f(x) = (x2 + 1), 4 x 24 0 Find the standard deviation of random variable X.
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Suppose the density function of a random variable X is 41 Find (1) coefficient A; (2) P0XI; (3) distribution function F(x). 7. Suppose X N(3,22) ( 1 ) Evaluate P {2-X 5), P {-4CK 10), (2) Decide C so that P {X > C }-P {XSCJ PIX >2)
1. Let X be a continuous random variable with probability density function f(x) = { if x > 2 otherwise 0 Check that f(-x) is indeed a probability density function. Find P(X > 5) and E[X]. 2. Let X be a continuous random variable with probability density function f(x) = = { SE otherwise where c is a constant. Find c, and E[X].
4. [10 pts] Let X be a random variable with probability density function if 1 < a < 2, 2 f(a)a 0 otherwise. Find E(log X). Note: Throughout this course, log = loge.
2. A random variable has a probability density function given by: Bmx-(B+1) x20 x<m fx(x)= 10 where m>0 and B > 2. Let m and ß be constants; answer the questions in terms of m and B. (a) Find the cumulative distribution function (cdf) Fx(x) of this random variable; (b) Find the mean of X; (c) Find E[X']; and (d) Find the variance of X. [12 points]
Let X be an exponential random variable such that P(X < 27) = P(X > 27). Calculate E[X|X > 23].
7. Let X, X, be a random sample with common pár 1 2 f(x) θ e-A, x > 0,0 > 0, 0 elsewhere. (a) Find the maximum likelihood estimator of θ, denoted by (b) Determine the sampling distribution of θ (c) Find Eô) and Var(). (d) What is the maximum value of the likelihood function? θ .
Problem 1. Let X be a contiuous random variable with probability density 2T f0SS Let A be the event that X > 1/2. Compute EXA) and Var(XA).
The probability density function for random variable Wis given as follows: 120 w>0 20 Let x be the 100pth percentile of W and y be the 100(1-p)th percentile of W, where o<p〈1. Express y as a function of x. ln(1-e- x/20 ) 20 -x/20 In 1-e 20 Cy- -20 ln 1-ex20 y20 n e-/20
6. Let X be a continuous random variable whose probability density function is: 0, x <0, x20.5 Find the median un the mode. 7. Let X be a continuous random variable whose cumulative distribution function is: F(x) = 0.1x, ja 0S$s10, Find 1) the densitv function of random variable U-12-X. 0, ja x<0, I, ja x>10.