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- A continuous random variable has probability density function given by K(2x-3);1<x2 Find k if f(x) is a probability density functionSuppose X is a random variable taking values in the interval [0,2] with probability density function f(x) = 1-x/2. What is the variance of X?A simple random sample X1, …, Xn is drawn from a population, and the quantities ln X1, …, ln Xn are plotted on a normal probability plot. The points approximately follow a straight line. True or false: a) X1, …, Xn come from a population that is approximately lognormal. b) X1, …, Xn come from a population that is approximately normal. c) ln X1, …, ln Xn come from a population that is approximately lognormal. d) ln X1, …, ln Xn come from a population that is approximately normal.
- Suppose that X and Y are independent and uniformly distributed random variables. Range for X is (−1, 1) and for Y is (0, 1). Define a new random variable U = XY, then find the probability density function of this new random variable.2.1) Suppose a random variable X has a probability density function in the attached image. a) Find the value of k that makes this a probability density function. b) Find the expected value of X.A random variable X has a Pareto distribution if andonly if its probability density is given by f(x) =⎧⎪⎨⎪⎩αxα+1 for x > 10 elsewhere where α > 0. Show that μ r exists only if r < α.
- A continuous random variable X has the probability density function f(x) so that f(x) is 0 for x≤≤≤≤0; f(x)=x for 0<x ≤ 1; f(x)= 2-x for 1<x≤ 2 and f(x) = 0 for x>2. Find the probability that X is at most 1/4.A continuous random Variable X has probability Density function defined by f(x) = 5-5x; 0Combined probability density function of continuous random variables X and Y -∞<x<∞ , -∞<y<∞ If it is assumed to be defined in the range, which of the following is / are true?
- 1)Independent random variables X and Y are exponentially distributed with mean is β . Which of the following is the expected value of the new random variable Z defined by Z = X + Y? 2)The composite probability density function for the random variables X and Y is given as f (x, y) = x + y, 0 <x <1, 0 <y <1, otherwise 0. Which of the following is the value of the probability P (X ≤ 1/2)?Let X1, X2, and X3 be independent random variables from (-1, 1). Find the probability density function and the expected value of the random variable [X(1) + X(2)]/2.Suppose that f(x) (as given above) is the probability density function of some random process (technically, of a continuous random variable). Suppose A = [−3, 3] ∪ [6, 12] is the event that the process (or random variable) returns a value between −3 and 3 or between 6 and 12, and that B = [2, 9] is the event that the process (or random variable) returns a value between 2 and 9. 2. Compute P(B|A) and P(A|B).