Question

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

Answer #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.

let the density function of x be f(x) = e^−x, x>0, find of
the density function of Z = e^-x

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

7. For the random variable x with probability density function:
f(x) = {1/2 if 0 < x< 1, x − 1 if 1 ≤ x < 2}
a. (4 points) Find the CDF function. b. (3 points) Find p(x <
1.5). c. (3 points) Find P(X<0.5 or X>1.5)

1. f is a probability density function for the random
variable X defined on the given interval. Find the
indicated probabilities.
f(x) = 1/36(9 − x2); [−3, 3]
(a) P(−1 ≤ X ≤ 1)(9 −
x2); [−3, 3]
(b) P(X ≤ 0)
(c) P(X > −1)
(d) P(X = 0)
2. Find the value of the constant k such that the
function is a probability density function on the indicated
interval.
f(x) = kx2; [0,
3]
k=

Find the value of C > 0 such that the function
?C sin2x, if0≤x≤π,
f(x) =
0, otherwise
is a probability density function.
Hint: Remember that sin2 x = 12 (1 − cos 2x).
2. Suppose that a continuous random variable X has probability
density function given by the above f(x), where C > 0 is the
value you computed in the previous exercise. Compute E(X).
Hint: Use integration by parts!
3. Compute E(cos(X)).
Hint: Use integration by substitution!

a) The joint probability density function of the random
variables X, Y is given as
f(x,y) =
8xy
if 0≤y≤x≤1 , and 0
elsewhere.
Find the marginal probability density functions.
b) Find the expected values EX and
EY for the density function above
c) find Cov X,Y .

Probability density function of the continuous random variable X
is given by f(x) = ( ce −1 8 x for x ≥ 0 0 elsewhere
(a) Determine the value of the constant c.
(b) Find P(X ≤ 36).
(c) Determine k such that P(X > k) = e −2 .

The probability density function for a continuous random
variable X is given by
f(x) =
0.6 0<X<1
=
0.10(x) 1 ≤X≤ 3
=
0 otherwise
Find the 85th percentile value of X.

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.

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