Let X be a random variable with p.d.f. x= 1,2,3,4,5 5 f(x) = , find E(x)

Q: 2. Suppose that X is a random variable with E(X) = 6 and Var(X) = 4. Find the following: a) Е(4X +…

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Q: Show that, if the random variable X has an F-distribution with ν1 and ν2 degrees of freedom, then Y…

A: If there exist two independent variables with degrees of freedom v1,v2 is the random variable of…

Q: If X₁, X₂, X3, .., Xn are random sample from a population with mean μ and variance o², then what is…

A: Given that X1,X2,...,Xn is a random sample from a population with mean μ and variance σ2 and…

Q: 7. If X is a random variable with fx Let Y = 3√In X. (a) Fx(x) ) = 1, 1<x<e (b) Fy(y) (c) fy(y)

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Q: Let x = (x1, x2 ..., xi, ..., xn) be a data set with a sample mean x̄ and a sample variance Sx2.…

A: Given  x=x1,x2,…,xny=y1,y2,…,ynx=x1+x2+....+xnny=y1+y2+....+ynn Given  z=x-y…

Q: Let X be a uniform random variable over the interval 1 to 4 and Y is exponential with a mean of 2.…

A: Given information: X~Uni(a=1,b=4) Standard deviation of X is: σX=b-a12       =4-112       =0.8660…

Q: 7. Suppose that X is a random variable for which the m.g.f. is as follows: V(1) = -(3e' + e!) for -0…

A: The moment generating function of the random variable X is given as, MX(t)=E(etx)=ψ(t) =143et+e-t…

Q: 8. Let X1, X2 be i.i.d. random variables from a distribution with p.d.f xe r>0 fx(x) = otherwise and…

A: The distribution is given by, fX(x) = xe-xx&gt;00Otherwise So,  P(X≥x)=∫x∞fX(t) dt=∫x∞te-t…

Q: (a) (b) Find E(W). Find Var(W).

A: Given,let  be  independent random variables.E(Yi)=μ and Var(Yi)=σ2  for every since (a)we want to…

Q: px (k) = c( (4)* k for k = 1,2,3,.. 3. Let X be a discrete random variable with %3D 3 A. Find the…

A: A) The value of c is calculated as follows: ∑1∞c13k=1c∑1∞13k=1 c13+132+133+...=1 c31+13+132+...=1…

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 x1  and x1 be independent standard normal random variables. If Y1= x1 + 2x2 and  Y2 = 2x1 - x2,…

A: We know, any linear combination of normal distribution is a normal distribution. Here, x1 and x2 are…

Q: Yı, Y2, ..., Yn are i.i.d. exponential random variables with E{Y;} = 1/0. Find the distribution of Y…

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

Q: Suppose that three random variables x,„X,„X, fom a random sample from the uniform distribution on…

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Q: 2. Let X be a random variable with p.m.f., (1+x? 20X = -1,0,1,2,3 0 ,otherwise. f(x) = Find E(3X² +…

A: Given function isf(x)={1+x220,    x=-1,0,1,2,30         ,   otherwise

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!