Question

Let X ∼ Uniform([a, b]).

a). Compute the cumulative distribution function (cdf) of X.

b). Prove that E(X) = (a + b)/2.

Answer #1

Compute the Cumulative Distribution function for the following
distributions knowing only f(x), E(x), and Var(x)
Uniform (over (a,b))
Bernoulli
Binomial
Geometric
Negative Binomial

Suppose a random variable X has cumulative distribution function
(cdf) F and probability
density function (pdf) f. Consider the random variable Y =
X?b
a for a > 0 and real b.
(a) Let G(x) = P(Y x) denote the cdf of Y . What is the
relationship between the functions
G and F? Explain your answer clearly.
(b) Let g(x) denote the pdf of Y . How are the two functions f
and g related?
Note: Here, Y is...

Let X have a uniform distribution on (0, 1) and let y = -ln ( x
)
a. Construct the CDF of Y graphically
b. Find the CDF of Y using CDF method
c. Find the PDF of Y using PDF method

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 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 ).

X is a continuous random variable with the cumulative
distribution function
F(x) = 0
when x <
0
= x2
when 0 ≤ x ≤
1
= 1
when x >
1
Compute P(1/4 < X ≤ 1/2)
What is f(x), the probability density function of X?
Compute E[X]

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).

Suppose that X is continuous random variable with PDF f(x) and
CDF F(x). (a) Prove that if f(x) > 0 only on a single (possibly
infinite) interval of the real numbers then F(x) is a strictly
increasing function of x over that interval. [Hint: Try proof by
contradiction]. (b) Under the conditions described in part (a),
find and identify the distribution of Y = F(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 .

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).

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