(33) Let X be a random variable with p.d.f. e2k x= 1,2,3 f(x)=- find (1) k (2) E(x+1) O.W

Q: (17) Let X be a random variable with p.d.f. o >x>0 f(x)=< find E(e) O.w

A: Solution : Given :  f(x) = 15e-15x          0&lt;x&lt;∞0           o.w Here X ~exp(15) M.G.F of x is…

Q: (52) Let X be a random variable with p.d.f. 1 5 0 <x<0 5 2 f(x) =- find E(e3 ) O.W

A: Given, X is a random variable with pdf- fx=15e-15x  ;0&lt;x&lt;∞0            ; otherwise

Q: Calculate E((X +Y)²).

A:

Q: 7- Suppose X, Y and Z be random variables with j.p.m.f. f(x, y, z)= kxyz with x E {1,2}, y E…

A: Given information: fx,y,z=kxyz, x=1,2, y=1,2,3 and z=1,2 The value of k is obtained below:…

Q: If the moment generating function of a random variable X is: (1/3+(2/3)e t ) 5 find P (X > 3).

A: The given moment generating function of a random variable X is 13+23et5.

Q: Consider the random variables X, and Y₁ from question B1 again. (a) Show that the sample covariance…

A: a) Recall that the sample covariance between X and Y is defined as: Cov(X,Y) = 1/n * ∑(Xi - x̄)(Yi -…

Q: (1) Let X be a random variable with p.d.f. ke 2x f(x) ={ 0 <x<0 , then o.w (a) k=3 (b) k=2 (c) k =-…

A: Reviewing information, X is a r.v. with pdf, f(x)=k·e-2x                                   ;…

Q: 2. Let X₁, X₂₁.... X ₁ be a i.i.de Sequence of random variables. Show that Cov (X₂X₁ X) = 0₁ V₂ =…

A:

Q: Find E (X) and E(X²) and using these values evaluate E[(2X + 1)²]

A:

Q: Find the distribution functions and con- ditional cdf's.

A:

Q: Let X denote a random variable that takes on any of the values -1, 0, 1 with respective…

A: P(X = −1) = 0.2,P(X = 0) = 0.5,P(X = 1) = 0.3.

Q: (V) Suppose A and Y are random variables with E. correlation coefficient of X and Y? -x+ 3 and what

A:

Q: (1) Let X be a random variable with a function x= 0,1, 2,3 4 xt(3-x)! S(x)=- , then E(x+1) is o.w…

A:

Q: 9. A random variable X has the following p.m.f: X0 1 2 3. P J0.1 0.3 0.4 0.2 And F(x) is the cdf for…

A: It is given that the, A random variable X has the pmf as X P(X=x) 0 0.1 1 0.3 2 0.4 3 0.2…

Q: 6x) 0<x<1 S(x)%=D , then p(x<1) is (a) 1 (b) 2 (c) 3 (d) 4 (c) None of these

A: It is an important part of statistics. It is widely used.

Q: (1) Let X be a random variable with a function [3 1 x= 0,1,2,3 4 x! (3-x)! fx) = { , then E(x) is…

A: Given, X is a random variable with the function below

Q: Suppose the p.d.f. of random variable X is defined as x³/20 if1<x< 3 f(x) = otherwise

A:

Q: 20) Let X be a random variable with p.d.f. k T<x<10 f(x) =- , find k. O.w

A: Solution: Let X be the random variable with pdf f(x) is  f(x)=kx       1&lt;x&lt;100      otherwise

Q: 13. The loss random variable X has a probability generating function 1 P(z) = = 2-z What is E[XA 3]?

A:

Q: 6. Suppose X is a random variable with E(X) = 4 and Var(X) = 9. Let Y = 4X + 5. Compute E(Y) and…

A: In question, Given E(X) = 4 and Var(X)=9 Y=4X+5 Then we'll find the value of E(Y). The solution is…

Q: 1. The sum of two exponential random variables: 1.1 The standard exponential random variable has R…

A: Let x be a continuous random variable that follows  a standard exponential distribution with the…

Q: Let X and Y be independent random variables, both being equally likely to be any of the values 1, 2,…

A: The  and  are independent random variables. and takes equally likely to be any of the values .

Q: Suppose X has probability generating function GX(t) = 0.2 + 0.3t + 0.1t2 + 0.4t3. What is P(X = 2)?…

A: PGF of a Random Variable is given by, Gx(t) = Σ tx P(X=x)

Q: 16 a) A random variable has pdf £x (x) = ² (1-x²) fx 4 0 (1-x²) 0≤x≤1 Find E (4x+2) and E(x²). else…

A: Given information: f(x)=541-x4      ; 0≤x≤10                     ; Elsewhere

Q: -. Determine the constant c so that f(x)=cx, x=1, 2, 3,...,10 satisfies the conditions of being a p.…

A: The given probability distribution can be desctibed as X 1 2 3 4 5 6 7 8 9 10 f(x) c 2c 3c 4c…

Q: A) Caused by a third variable B) uncorrelated ) negative effect 0.fb/ orrrte D) direct…

A: The correlation coefficient formula is given by

Q: 1. Let X1, X2, X3 be identically distributed independent exponential random variables and Y = X₁+ X2…

A: Given that X​1, X2​, and X3​ are identically distributed independent exponential random variables…

Q: le covariance is an unbiased estimator of Cov(X₁, Y₁). equality X₁ X = (Xi − µ) — (X − µμ). - - - om…

A: Step 1: proove part (a)a)Recall that the sample covariance between X and Y is defined as:Cov(X,Y) =…

Q: A student was asked to find a 90% confidence interval for the proportion of students who take notes…

A: Given,sample size(n)=81confidence interval(C.I)=0.15&lt;p&lt;0.31α=1-0.90=0.1α2=0.05Z0.05=1.645…

Q: 7. A discrete random variable X taking non-negative integer values has probability gen- erating…

A:

Q: Hom variable X has the gamma p. d. f with integer j rary ß>0, then E(Xk) is

A:

Q: (7) Let X be a random variable with p.d.f. 1<x<0 f(x) = { ,find k. O.W

A: Reviewing information, X is a r.v. with pdf,f(x)=kx-2                          1&lt;x&lt;∞0…

Q: 1) Assume a relationship Y = g2(91(X)) between the two random variables X and Y where gi and g2 are…

A: @solution::::

Q: 10- If the random variable X has the gamma p. d. f with integer parameter a and arbitrary ß> 0, then…

A:

Q: 3.2.13 Y is a continuous random variable with { 2(1- y), 0< y < 1; fy(4) = { 0, otherwise. Derive…

A: Let Y denotes the Continuous Random variable. From the information, given that

Q: 14. Let X be a discrete random variable with the following PMF 1 for k = -2 4 1 for k = -1 Px (k) 1…

A: Farmula used  Range=Maximum-Minimum

Q: Let A be the mean of the random sample drawn from a population whose pdf is given by fox =…

A: Note: Hi, thank you for the question. As per our company guideline we are supposed to answer only…

Q: 1.1 Consider Y₁, Y2,...,Yn are i.i.d. random variables with n > 4 and T is the sample mean Y. Also…

A: From the given information, Y1, Y2,…………..,Yn are i.i.d random variables.

Q: Find the marginal pdf's

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!