5. Let X and Y be independent random variables such that X - exp(A) and Y ~ exp(). (a) What is the distribution of X +Y? (b) What is the conditional pdf of X, given that X +Y = 1?

Q: Let X be a binomial random variable with probability distribution 3-a SOOʻO**, x = 0,1, 2, 3, f(x) =…

A: Solution-: Given: n=3,p=2/5 ∴X→B(n=3,p=2/5) The p.m.f. of X is given by,  P(X=x)=3x*(25)x*(35)3-x…

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: Exercise 23. Let X be a random variable having the uniform distribution on the interval (0,0+ 1), 0…

A:

Q: 63. Let the random variables X~ N(ux, o) and Y~ N(μy, o) be jointly continuous normal random…

A: Since here:

Q: Q1) Let be iid Poisson(2.), let à. be a r(a, ß) distribution, the conjugate family for the Poisson.…

A: Given that x/λ~PλthenL(x/λ)=∏i=1nf(xi,λ)=∏i=1ne-λλxixi!=e-nλλΣxi∏i=1nxi!λ~α,βf(λ)=e-βλλα-1βαα

Q: If 'b' is Constant and X be a Random Variable then VX +b)3V(X)

A:

Q: Let X1,.. X„ be n independent random variables with PDF fx, (x) = - [A,e- if x > 0, i = 1,2, ..., n.…

A:

Q: d) The random variable X has a Poisson distribution such that P(X= 0) = P(X = 1). Calculate P(X =…

A: Given problem Given that   d)   The random variable X has a Poisson distribution such that P( X=0…

Q: If X~beta(3,2) is independent of Y~beta(3,4), what is the distribution of X+Y

A: The objective of this question is to find the distribution of the sum of two independent random…

Q: 3. (15 points) Let X1 and X2be independent exponential random variables: fx, (x1) = e-#1 and fx,(x2)…

A: # Random variable x1 & X2 are independent exponential random variable  with pdf f(x1)=e^-x1…

Q: 2. A random variable X follows Beta(2,3) distribution. (a) Show that the cumulative distribution…

A: Here, Beta(2,3) F(x) = 6x^2 - 8x^3 + 3x^4

Q: Let X₁,..., Xn be iid Poisson(A), and let λ have a gamma(a, 3) distribution, the conjugate family…

A:

Q: Q5. A) Consider a sample X,,X,*,X, of normally distributed random variables with variance o= 11.…

A: Note: As per Bartleby’s guideline for more than one questions only first is to be answered please…

Q: A random variable Y has the Gaussian distribution on R with mean a and variance o?, Select all the…

A: a and c.

Q: Suppose that Y1,..., Yn is a sample of size n from a Exp(B), B > 0. .3B (a) Find the distribution of…

A:

Q: Let X₁, X2, ..., XK-1 have a multinomial distribution. (a) Find the mgf of X₁, X₂, ..., XK-2. (b)…

A:

Q: Let X be a random variable with p.d.f f(x) =)",x = 1,2,3,. Find P(A), where A={1,3,5, ..} 2/3 2/5 O…

A:

Q: n F. De X(w) i- -X(w)

A: the distribution function is also known as a cumulative distribution function.shows that the…

Q: Let X~ N(0, 1) and let Y = -X. (a) Show that Y~ N(0, 1) (by showing that Y has cdf Þ). (b) What is…

A: X follows a standard normal distribution.

Q: Let T t(7), then the distribution of T is distribution.

A: Given,  T~t(7)

Q: Let X andY be independent I'(2, a)-distributed random variables. Find the conditional distribution…

A: Introduction: The random variables X and Y are independent of each other, and each follows the gamma…

Q: Suppose random variable X has a Poisson distribution with t​ = 2 and λ= 4. What is​ E(X)?

A:

Q: 3. Let Y be the number of speeding tickets a YSU student got last year. Suppose Y has probability…

A:

Q: 8) Let X be a random variable, and let g,(x) and 92(x) any two functions whose expectations exist,…

A: Let X be a random variable and let g1x and g2x any two functions whose expectations exist    where…

Q: Suppose that the Cumulative distribution function (CDF) of the random variable X with parameter 0 >…

A:

Q: A random point (X,Y) is distributed uniformly on the square with vertices (1,1), = on the square.…

A:

Q: 2. Let Y₁,..., Yp be independent random variables such that Y~ N(0, 1). Write Y = (Y₁...., Yp) and 0…

A: To show that an unbiased estimator of the risk of the James-Stein estimator iswe can use the…

Q: Let X ~ N(0,1), the standard Normal distribution. (a) Determine the 3rd moment of X. (b) Determine…

A:

Q: When do you call two random processes to be joinily wide-sense stationary ?

A:

Q: and let Y; = Xí, i = 1, ..., n. %3D Find the distribution of W 2(Y1+Y2+Y3) 3(Y4+Y5)

A:

Q: Let the probability distribution of random variable X be P(X= k) k! , k = 0, 1, 2, I then EX2 - ()

A: Let X has a pmf

Q: Y)~ N₂(4, 5, 9, 4,-0.5). What is the conditional distribution of Y given X = 1?

A: Given that  (X,Y)~ N2(4,5,9,4,-0.5) Where μ1=4 , μ2=5 ,?12=9 , ?22=4 And ρ= -0.5 So follow normal…

Q: (2) (0,8)" (0,2)³-* si x = 0,1, 2, 3 fx(x) = in another case

A: Formula used.  In binomial distribution.  Expected value = E(x)= mean  E(x)= np And  Variance (x)=…

Q: Let Y~Po(1) be a poisson distributed variable with parameter 2 = 1. Suppose that, designated on Y, X…

A:

Q: Find Cumulative Distribution Function (cdf) of X, F(x), x {0, 1, 2,...}

A: The cumulative distribution function is: F(a)=∑x=0ac13x=c∑x=0a13x=c1301-13a1-13=c(1)1-13a23=3c21-13a…

Q: Suppose that X and Y are jointly distributed random variables such that E(YX)=a+bX. (1) Express a…

A:

Q: E estimator c onal distribut us part, find

A: Hi! Thank you for the question, As per the honor code, we are allowed to answer three sub-parts at a…

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!