2. Let X be an exponentially distributed random variable with parameter 1 = 2. Determine P(X...
Let X be an exponentially distributed random variable with parameter λ. Prove that P(X > s + tK > t) P(X > s) for any S,12 0
Let h be an exponentially-distributed random variable with the distribution function p- exp(-x) for x > 0 and ph = nction Ph 0 for a s 0. Derive the distribution function of its square root, Solution: 2y exp(-y2
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.
2x 0<x<1 Let X be a continuous random variable with probability density function f(x)= To else The cumulative distribution function is F(x). Find EX.
Let X be an exponentially distributed random variable with parameter value 2. a) Use the markov inequality to estimate P(X >= 3). b) Use the chebyshev inequaity to estimate P(X >= 3). c) Calculate P(X >= 3) exactly.
PLEASE ANSWER ALL QUESTION 1 1 points Save Answer A random variable is a uniform random variable between 0 and 8. The probability density is 1/8, when 0<x<8 and O elsewhere. What is the probability that the random variable has a value greater than 2? QUESTION 2 1 points Save Answer The total area under a probability density curve of a continuous random variable is QUESTION 3 1 points Save Answer X is a continuous random variable with probability density...
2. Let X be a continuous random variable with pdf f(x) = { cr", [w] <1, f() = 0. Otherwise, where the parameter c is constant (with respect to x). (a) Find the constant c. (b) Compute the cumulative distribution function F(2) of X. (c) Use F(2) (from b) to determine P(X > 1/2). (d) Find E(X) and V(X).
4. (30pts) A continuous random variable X has the probability density function: hx - 1 sx 32 f(x) =Jo-hx 2 x 3 0 x >3 which ean bo graphed as f(x) 1 2 a) Find h which makes f(x) a valid probability density function b) Find the expected value E(X) of the probability density function f(x) c) Find the cumulative distribution function F(x). Show all you work
Let X be exponentially distributed with parameter 3. a) Compute P(X > 6 | X > 2). b) Compute E(7e-12x+8+ 5). c) Let Y be independent from X. Suppose the PDF for Y is f(x) = 2x for 0 ≤ x ≤ 1 (and 0 else). Find the PDF of X + Y.
2. Let X be a continuous random variable with pdf ( cx?, [xl < 1, f(x) = { 10, otherwise, where the parameter c is constant (with respect to x). (a) Find the constant c. (b) Compute the cumulative distribution function F(x) of X. (c) Use F(x) (from b) to determine P(X > 1/2). (d) Find E(X) and V(X).