Question

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

Answer #1

i) Uniform distribution:

ii) Bernoulli distribution:

iii) Binomial Distribution

4) Geometric:

5) Negative Binomial distribution:

Let X ∼ Uniform([a, b]).
a). Compute the cumulative distribution function (cdf) of X.
b). Prove that E(X) = (a + b)/2.

If X is a uniform distribution defined over the interval (a,b),
verify that
E(X)=(a+b)/2
Var(X)= (b-a)2/12 (Hint:Use the computing
formula)

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]

a) let X follow the probability density function f(x):=e^(-x) if
x>0, if Y is an independent random variable following an
identical distribution f(x):=e^(-x) if x>0, calculate the moment
generating function of 2X+3Y
b) If X follows a bernoulli(0.5), and Y follows a
Binomial(3,0.5), and if X and Y are independent, calculate the
probability P(X+Y=3) and P(X=0|X+Y=3)

The cumulative distribution function for a random variable X is
F(x)= 0 if x less than or equal 0, or F(x)=sinx if 0 is less than x
is less than or equal to pi/2 , or F(x)= 1, if x is greater than
pi/2 . (a) find P(0.1 less than X less than 0.2. (b) find E(x)

Suppose that the random variable X has the following cumulative
probability distribution
X: 0 1. 2. 3. 4
F(X): 0.1 0.29. 0.49. 0.8. 1.0
Part 1: Find P open parentheses 1 less or equal than
x less or equal than 2 close parentheses
Part 2: Determine the density function f(x).
Part 3: Find E(X).
Part 4: Find Var(X).
Part 5: Suppose Y = 2X - 3, for all of X, determine
E(Y) and Var(Y)

Suppose the reaction temperature X (in °C) in a certain chemical
process has a uniform distribution with a= -5 and b = 5.
a) Compute P(X < 0)
b) Compute P(-2.5 < X < 2.5)
c) Find F(x), E(X), and var(X).

Show the following:
a) Let there be Y with the cumulative distribution function
F(y). Let F(Y)=Z. Show that Z~U(0,1) for F(y).
b) Let X~U(0,1), and let Y := -ln(X). Show that Y~exp(1)

The density function of random variable X is given by f(x) = 1/4
, if 0
Find P(x>2)
Find the expected value of X, E(X).
Find variance of X, Var(X).
Let F(X) be cumulative distribution function of X. Find
F(3/2)

Find E(X),Var(X),σ(X) if a random variable x is given by its
density function f(x), such that f(x)=0, if x≤0 f(x)=2x5, if
0<x≤1 f(x)=0, if x>1

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