X ~ Exp(
= 4)
The PDF of X is,
for x > 0
Y | X = x ~ N(
= 0,
= x)
Using Normal distribution equation and substituting
= 0,
= x, we get
the conditional probability density of Y given X = x as,
The first option is the correct answer.
Suppose that X is exponentially distributed with parameter 1 = 4. Given that X=X, Y is...
Suppose that X is exponentially distributed with parameter 1 = 4. Given that X=X, Y is normally distributed with mean and standard deviation x. Which of the following is the conditional probability density of Y given X=x? (1/a)2 e 2 72 73 e 2 √2 ay (y/-)2 4e 43 e 2 72 73 4 e 40 e 2 V2 πμ
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.
Suppose that is exponential distributed with parameter - 3. Given that is normally distributed with mean and standard deviation x. Which of the following is the concitional probebidensy of given Xx?
(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
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.
Suppose that the time (in hours) required to repair a machine is an exponentially distributed random variable with parameter λ (lambda) = 0.5.What's the probability that a repair takes less than 5 hours? AND what's the conditional probability that a repair takes at least 11 hours, given that it takes more than 8 hours?
Question 3 [17 marks] The random variable X is distributed exponentially with parameter A i.e. X~ Exp(A), so that its probability density function (pdf) of X is SO e /A fx(x) | 0, (2) (a) Let Y log(X. When A = 1, (i) Show that the pdf of Y is fr(y) = e (u+e-") (ii) Derive the moment generating function of Y, My(t), and give the values of t such that My(t) is well defined. (b) Suppose that Xi, i...
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.
Suppose that X is exponentially distributed with mean 1/2. Which of the following is a density function of Y = X? S4ye -2y2 0 y > 0 y < 0 2e-2y y > 0 y < 0 o {:2 2e-2y y > 0 y<0 Site 10 y > 0 y<0 ke-u/2 y y > 0 y < 0 { e-2y2 y > 0 y < 0 0
Suppose that X is exponentially distributed with mean 1/2. Which of the following is a density function of Y = X? S4ye -2y2 0 y > 0 y < 0 2e-2y y > 0 y < 0 o {:2 2e-2y y > 0 y<0 Site 10 y > 0 y<0 ke-u/2 y y > 0 y < 0 { e-2y2 y > 0 y < 0 0