Question

Let X be a random variable with probability density function fX(x) = {c(1−x^2)if −1< x <1, 0 otherwise}.

a) What is the value of c?

b) What is the cumulative distribution function of X?

c) Compute E(X) and Var(X).

Answer #1

Let X be a random variable with probability density function
fX(x) given by fX(x) = c(4 − x ^2 ) for |x| ≤ 2 and zero
otherwise.
Evaluate the constant c, and compute the cumulative distribution
function.
Let X be the random variable. Compute the following
probabilities.
a. Prob(X < 1)
b. Prob(X > 1/2)
c. Prob(X < 1|X > 1/2).

Let X be a random variable with the probability density function
fx(x) given by:
fx(x)=
1/4(2-x), 0<x<2
1/4(x-2), 2<=x<4
0, otherwise.
Let Y=|X-3|. Compute the probability density function of Y.

Let X and Y be a random variables with the joint probability
density function fX,Y (x, y) = { e −x−y , 0 < x, y < ∞ 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 .

Let the probability density function of the random variable X be
f(x) = { e ^2x if x ≤ 0 ;1 /x ^2 if x ≥ 2 ; 0 otherwise}
Find the cumulative distribution function (cdf) of X.

Let X be a random variable with probability density function
f(x) = { λe^(−λx) 0 ≤ x < ∞
0 otherwise } for some λ > 0.
a. Compute the cumulative distribution function F(x), where F(x)
= Prob(X < x) viewed as a function of x.
b. The α-percentile of a random variable is the number mα such
that F(mα) = α, where α ∈ (0, 1). Compute the α-percentile of the
random variable X. The value of mα will...

Let X be a continuous random variable with probability density
function (pdf) ?(?) = ??^3, 0 < ? < 2.
(a) Find the constant c.
(b) Find the cumulative distribution function (CDF) of X.
(c) Find P(X < 0.5), and P(X > 1.0).
(d) Find E(X), Var(X) and E(X5 ).

Let X be a random variable with probability density function fX
(x) = I (0, 1) (x). Determine the probability density function of Y
= 3X + 1 and the density function of probability of Z = - log
(X).

If the probability density function of a random variable X is
ce−5∣x∣ , then (a) Compute the value of c. (b) What is the
probability that 2 < X ≤ 3? (c) What is the probability that X
> 0? (d) What is the probability that ∣X∣ < 1? (e) What is
the cumulative distribution function of X? (f) Compute the density
function of X3 . (g) Compute the density function of X2 .

A continuous random variable X has the following
probability density function F(x) = cx^3, 0<x<2 and 0
otherwise
(a) Find the value c such that f(x) is indeed
a density function.
(b) Write out the cumulative distribution function of
X.
(c) P(1 < X < 3) =?
(d) Write out the mean and variance of X.
(e) Let Y be another continuous random variable such
that when 0 < X < 2, and 0 otherwise. Calculate
the mean of Y.

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).

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