
X is a random variable exponentially distributed with mean Y, where Y is uniformly distributed on...
Let X be uniformly distributed in the unit interval [0,2]. Consider the random variable Y=g(x), where g(x)=2, if x ≤ 1/4 and g(x) = 3, if X > 1/4. a.) find the expected value of Y (by first deriving its PDF)
If X is uniformly distributed over (0,2) and Y is exponentially distributed with parameter λ = 2. Also X and Y are independent, find the PDF of Z = X+Y.
Assume random variable ? is uniformly distributed in the
interval (−?/2 ,?⁄ 2]. Define the random variable ?=tan (?), where
tan (∙) denotes the tangent function. Note that the derivative of
tan (?) is 1/(cos (?)2) .
a) Find the PDF of ?.
b) Find the mean of ?
.Define the random variable ?=1/?.
c) Find the PDF of ?.
Assume random variable X is uniformly distributed in the interval (-1/2, 1/2). Define the random variable Y = tan(X), where...
Let X be a random variable with PDF fx(X). Let Y be a random variable where Y=2|X|. Find the PDF of Y, fy(y) if X is uniformly distributed in the interval [−1, 2]
(iv) Let X be exponentially distributed with parameter 1 and let Y be uniformly distributed in the interval [0, 1]. Using convolution, find the probability distribution function of
2. Let X and Y be independent, exponentially distributed random variables where X has mean 1/λ and Y has mean 1/μ. (a) What is the joint p.d.f of X and Y? (b) Set up a double integral for determining Pt <X <Y) (c) Evaluate the above integral. (d) Which of the following equations true, and which are false? {Z > t} = {X > t, Y > t} (e) Compute P[Z> t) wheret 0. (f) Compute the p.d.f. of Z.
Suppose that a random variable X is exponentially distributed with a mean of 13.6. Find the standard deviation of the distribution of the sample mean of a random sample of size 37. Round your answer to four decimal places.
2. Let X and Y be independent, exponentially distributed random variables where X has mean 1/λ and Y has mean 11. (a) What is the joint p.d.f of X and Y? (b) Set up a double integral for determining Pt < X <Y). (c) Evaluate the above integral. (d) Which of the following equations true, and which are false? (e) Compute PIZ> t where t20. (f) Compute the pd.f. of Z. Z = min(X,Y)
3. The pair of random variables X and Y is uniformly distributed on the interior of the triangle with the vertices whose coordinates are (0,0), (0,2), and (2,0) (i.e., the joint density is equal to a constant inside the triangle and zero outside). (a) (10 points) Find P(Y+X< 1). (b) (10 points) Find P(X = Y). (c) (10 points) Find P(Y > 1X = 1/2).
3. Suppose that X and Y are independent exponentially distributed random variables with parameter λ, and further suppose that U is a uniformly distributed random variable between 0 and 1 that is independent from X and Y. Calculate Pr(X<U< Y) and estimate numerically (based on a visual plot, for example) the value of λ that maximizes this probability.