Question

Let X1,...,X99 be independent random variables, each one distributed uniformly on [0, 1]. Let Y denote the 50th largest among the 99 numbers. Find the probability density function of Y.

Answer #1

Answer:-

Let X1, X2, X3 be independent random variables, uniformly
distributed on [0,1]. Let Y be the median of X1, X2, X3 (that is
the middle of the three values). Find the conditional CDF of X1,
given the event Y = 1/2. Under this conditional distribution, is X1
continuous? Discrete?

Let X and Y be independent; each is uniformly distributed on [0,
1]. Let Z = X + Y. Find:
E[Z|X]. Your answer should be a function of x.

Let Xi, i = 1, 2..., 48, be independent random variables that
are uniformly distributed on the interval [-0.5, 0.5].
(a) Find the probability Pr(|X1|) < 0.05
(b) Find the approximate probability P (|Xbar| ≤ 0.05).
(c) Determine an approximation of a such that P(Xbar ≤ a) =
0.15

Suppose that X1 and X2 are independent continuous random
variables with the same probability density function as: f(x) = ( x
2 0 < x < 2, 0 otherwise. Let a new random variable be Y =
min(X1, X2,).
a) Use distribution function method to find the probability
density function of Y, fY (y).
b) Compute P(Y > 1).

Let Y be the liner combination of the independent random
variables X1 and X2 where Y = X1 -2X2
suppose X1 is normally distributed with mean 1 and standard
devation 2
also suppose the X2 is normally distributed with mean 0 also
standard devation 1
find P(Y>=1) ?

Let {Xj} ∞ j=1 be a collection of i.i.d. random variables
uniformly distributed on [0, 1]. Let N be a Poisson random variable
with mean n, and consider the random points {X1 , . . . , XN }.
b. Let 0 < a < b < 1. Let C(a,b) be the number of the
points {X1 , . . . , XN } that lies in (a, b). Find the conditional
mass function of C(a,b) given that N =...

Suppose that X1 and X2 are independent continuous random
variables with the same probability density function as: f(x) = ( x
2 0 < x < 2, 0 otherwise. Let a new random variable be Y =
min(X1, X2,).
a) Use distribution function method to find the probability
density function of Y, fY (y).
b) Compute P(Y > 1).
c) Compute E(Y )

Let X1, X2 be two normal random variables each with population
mean µ and population variance σ2. Let σ12 denote the covariance
between X1 and X2 and let ¯ X denote the sample mean of X1 and X2.
(a) List the condition that needs to be satisﬁed in order for ¯ X
to be an unbiased estimate of µ. (b) [3] As carefully as you can,
without skipping steps, show that both X1 and ¯ X are unbiased
estimators of...

Let X and Y be independent random variables, uniformly
distribued on the interval [0, 2]. Find E[e^(X+Y) ].

Let X1 and X2 be two independent geometric
random variables with the probability of success 0 < p < 1.
Find the joint probability mass function of (Y1,
Y2) with its support, where Y1 =
X1 + X2 and Y2 =
X2.

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