Question

How to find the mean of the probability density:

f(x) = e^-2x for -.441<x<.440, and 0 elsewhere.

Answer #1

1. Find k so that f(x) is a probability density function. k=
___________
f(x)= { 7k/x^5 0 1 < x < infinity elsewhere
2. The probability density function of X is f(x).
F(1.5)=___________
f(x) = {(1/2)x^3 - (3/8)x^2 0 0 < x < 2
elsewhere
3. F(x) is the distribution function of X. Find the probability
density function of X. Give your answer as a piecewise
function.
F(x) = {3x^2 - 2x^3 0 0<x<1 elsewhere

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 F(x) = 1 − e −2x for x > 0 and F(x) = 0 for x ≤ 0. Is
F(x) a distribution function? Explain your answer. If it is a
distribution function, find its density function.

Let X have the distribution that has the following probability
density function:
f(x)={2x,0<x<1
{0, Otherwise
Find the probability that X>0.5.
Why is the probability 0.75 and not 0.5?

Find the standard deviation of the distribution that has the
following probability density function:
f(x)={ 2x, 0<x<1 0, O.W.

Let the probability density of X be given by f(x) = c(4x - 2x^2
), 0 < x < 2; 0, otherwise. a) What is the value of c? b)
What is the cumulative distribution function of X?
c) Find P(X<1|(1/2)<X<(3/2)).

The probability density function of X is given by
f(x)={a+bx0if 0<x<1otherwise
If E(X)=1.5, find a+b.
Hint: For a probability density function f(x), we have
∫∞−∞f(x)dx=1

The random variable X has a probability density function f(x) =
e^(−x) for x > 0. If a > 0 and A is the event that X > a,
find f XIA (xlx > a), i.e. the density of the conditional
distribution of X given that X > a.

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)

Let X be a continuous random variable with the following
probability density function:
f(x) = e^−(x−1) for x ≥ 1; 0 elsewhere
(i) Find P(0.5 < X < 2).
(ii) Find the value such that random variable X exceeds it 50%
of the time. This value is called the median of the random variable
X.

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