Question
In: Math
The probability density function of the random variable X is given by fX(x) = ax +...
The probability density function of the random variable X is given by fX(x) = ax + 2/9 if 1/2 ≤ x ≤ 3, and 0 otherwise.
(a) Compute the value of a.
(b) Let the random variable Y be defined as Y = [X], where [·] is the “round down” operator (that is, for example, [2.5] = 2, [−2.5] = −3, [−3] = −3). Find the probability mass function of Y . (Hint: For Y to take value k, what values should X take?)
(c) Compute Var(Y )
I am confused with part B.
Solutions
milcah answered 1 year agoRelated Solutions
1. Let X be a random variable with probability density function fX given by fX(x) =...
1. Let X be a random variable with probability density function
fX given by fX(x) = γαγ/ (x + α)^γ+1 , x ≥ 0,
0, x < 0,
where α > 0 and γ > 0.
(a) Find the cumulative distribution function (cdf) FX of X.
(b) Let Y = log(X+α /α) . Find the cdf of Y and identify the
distribution.
(c) How could a realisation of X be generated from an R(0,1)
random number generator?
(d) Let Z...
The probability density function of the continuous random variable X is given by fX (x) =...
The probability density function of the continuous random
variable X is given by
fX (x) = kx, (0 <= x <2)
= k (4-x), (2 <= x <4)
= 0, (otherwise)
1) Find the value of k
2)Find the mean of m
3)Find the Dispersion σ²
4)Find the value of Cumulative distribution function FX(x)
The (mixed) random variable X has probability density function (pdf) fX (x) given by: fx(x)=0.5δ(x−3)+ {...
The (mixed) random variable X has probability density function
(pdf) fX (x) given by:
fx(x)=0.5δ(x−3)+ { c.(4-x2), 0≤x≤2
0, otherwise
where c is a constant.
(a) Sketch fX (x) and find the constant c.
(b) Find P (X > 1).
(c) Suppose that somebody tells you {X > 1} occurred. Find
the conditional pdf fX|{X>1}(x), the pdf of X given
that {X > 1}.
(d) Find FX(x), the cumulative distribution function of X.
(e) Let Y = X2 . Find...
Let X be a continuous random variable with a probability density function fX (x) = 2xI...
Let X be a continuous random variable with a probability density
function fX (x) = 2xI (0,1) (x) and let it be the function´ Y (x) =
e −x
a. Find the expression for the probability density function fY
(y).
b. Find the domain of the probability density function fY
(y).
[25 pts] The probability density function of a random variable X is fX(x) = ce^ -|x|...
[25 pts] The probability density function of a random variable X
is fX(x) = ce^ -|x| for all values of x, where c is a constant.
Find the value of c and the cumulative distribution function of
X.
If a random variable X has a beta distribution, its probability density function is fX (x)...
If a random variable X has a beta distribution, its probability
density function is
fX (x) = 1 xα−1(1 − x)β−1 B(α,β)
for x between 0 and 1 inclusive. The pdf is zero outside of
[0,1]. The B() in the denominator is the beta function, given by
beta(a,b) in R.
Write your own version of dbeta() using the beta pdf formula
given above. Call your function mydbeta(). Your function can be
simpler than dbeta(): use only three arguments (x, shape1,...
If a random variable X has a beta distribution, its probability density function is fX (x)...
If a random variable X has a beta distribution, its probability
density function is
fX (x) = 1 xα−1(1 − x)β−1 B(α,β)
for x between 0 and 1 inclusive. The pdf is zero outside of
[0,1]. The B() in the denominator is the beta function, given by
beta(a,b) in R.
Write your own version of dbeta() using the beta pdf formula
given above. Call your function mydbeta(). Your function can be
simpler than dbeta(): use only three arguments (x, shape1,...
If a random variable X has a beta distribution, its probability density function is fX (x)...
If a random variable X has a beta distribution, its probability
density function is
fX (x) = 1 xα−1(1 − x)β−1 B(α,β)
for x between 0 and 1 inclusive. The pdf is zero outside of
[0,1]. The B() in the denominator is the beta function, given by
beta(a,b) in R.
Write your own version of dbeta() using the beta pdf formula
given above. Call your function mydbeta(). Your function can be
simpler than dbeta(): use only three arguments (x, shape1,...
The joint probability density function for two random variables X and Y is given as, fx,y...
The joint probability density function for two random variables
X and Y is given as, fx,y (x, y) = (2/3)(1 + 2xy3 ), 0
< x < 1, 0 < y < 1
(a) Find the marginal probability density functions for X and Y
.
(b) Are X and Y independent? Justify your answer.
(c) Show that E[X] = 4/9 and E[Y ] = 7/15 .
(d) Calculate Cov(X, Y )
. Let X and Y be a random variables with the joint probability density function fX,Y...
. Let X and Y be a random variables with the joint probability
density function fX,Y (x, y) = { 1, 0 < x, y < 1 0, otherwise
} . a. Let W = max(X, Y ) Compute the probability density function
of W. b. Let U = min(X, Y ) Compute the probability density
function of U. c. Compute the probability density function of X + Y
..
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