The joint PDF of two random variables X,Y is given by fXY(x, y) = {k e^(-y-x/4) , x > 0, y > 0                     0,                  otherwise a)  What is the value of k ? b)  Find the probability P( X

Q: Let X1, . . . , Xn, . . . be iid random variables with common PDF f(x | θ) = e^−(x−θ) x > θ, 0 x ≤…

A:

Q: Prove that the variance of X ∼ Expo(λ) is 1/λ^2 by doing the calculation. You canuse the result E(X)…

A: To prove that the variance of an exponentially distributed random variable ( X ) with rate parameter…

Q: Random Variable X has PMF given by the following. What is the PMF of Y = X^2  + 1.   x…

A: Given Y=X2+1 X -1 1 2 P(X=x) 0.4 0.1 0.5

Q: 1 V(X) for i) X ~ U{1,n}, ii) X~ Geometric (p), iii) X P

A: We have given that X be a random variable which follows 1) Uniform distribution, i.e. X∼U{1,n}2)…

Q: If X is Bin(200, 0.5) (1) What is the expectation and variance of X? (2) Which approximation is…

A: Binomial Distribution: If X is a discrete random variable and it is said to follow Binomial…

Q: Suppose that the lifetime, X, and brightness, Y, of a light bulb are modeled as continuous random…

A: Joint Probability Distribution: Let X and Y be two continuous random variables defined over the…

Q: Suppose we have random samples (X₁, X2, X3, X4, X5) and it is known that X; ~ Exponential(0). `…

A: Given that Xi∼Exponential(θ) for all i=1,2,3,4,5. where θ is the rate parameter and θ>0The…

Q: how would you calculate a-d of question 2?

A: We will calculate one by one in the following manner from the given data:

Q: Let U1, U2, ... be i.i.d. Uni(0, 1) random variables. - log U; is distributed as exp(1). (2) Find…

A:

Q: 2. Let X1, X2,..., Xn be independent random variables having the same probability mass function f…

A: Given Probability mass function a) Finding the method – of moment estimator

Q: The time that a butterfly lives after emerging from its chrysalis can be modelled by a random…

A: Given that the random variable T has the probability distribution, PT>t=366+t2 for t≥0. The…

Q: Suppose X₁, X2,..., X₁ are i.i.d. exponential random variables with pdf given by fx(x) = [e-Az if x…

A: It is given that  are i.i.d. exponential random variables with pdf given as:

Q: If P(B) = ³/ and P(A'^B)= 1, for two events A and B, find P(A/B).

A:

Q: b) Let U1, U2 be a random sample of size 2 from a uniform distribution over the interval (0, I). Let…

A: Let U1,U2 be a random sample from a uniform distribution (0.1) so the density function is given as…

Q: Let X1, X2, X3 be 3 independent random variables that fit the Poisson distribution with A k…

A:

Q: Let X,Y be two random variables and take values in {0,1} and whose joint distribution is given by: (…

A: First, we need to calculate the expected values of X, Y, X^2, and Y^2. The expected value of a…

Q: Let X be a random variable with pdff(x) = 4x^3 if 0 < x < 1 and zero otherwise. Use the cumulative…

A: The PDF for following function is: 1). Y=X4 PDF=∫01Ydx Solving it:  PDF=∫01X4dxPDF=x5501PDF=15 2).…

Q: You have two random variables M and N where the joint pdf is given by  f_M,N(m,n) = b if m + n >=2,…

A: The region in which f=b is shown in figure:  f=0 for other values of (m,n)

Q: If null hypothesis is false because a drop has occurred, then the observed z will be?

A: It is given that the null hypothesis false. That is, it should be rejected.

Q: f(x, y): ) = {K(x² + x²) [K(x² + y²) 20 ≤ x ≤ 30, 20 ≤ y ≤ 30 otherwise (a) Determine the…

A: Given:  The front tire on a particular type of vehicle is filled with the pressure of 26 psi .…

Q: Suppose that two independent and identically distributed random variables X and Y have common mgf…

A: Two independent and identically distributed random variables X and Y have common mgf:- MX(t) = MY(t)…

Q: If X ∼ Gamma(2, 5), find E(X^2 ) andE(X^4 )

A: Solution: Let X follows gamma distribution with the parameters 2 and 5.

Q: Suppose Y1,.…,Yn are independent random variables with probability mass function f(y) = p"(1 –…

A:

Q: Suppose that X is a random variable for which the moment generating function is as follows: MX(t) =…

A: Hello! As you have posted 4 different questions, we are answering the first question. In case you…

Continuous Probability Distributions

Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.

Normal Distribution

Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!