Question

2. Let the probability density function (pdf) of random variable X be given by:

f(x) = C (2x - x²), for 0< x < 2,

f(x) = 0, otherwise

- Find the value of C. (5points)
- Find cumulative probability function F(x) (5points)
- Find P (0 < X < 1), P (1< X < 2), P (2 < X <3) (3points)
- Find the mean, : , and variance, F². (6points)

Answer #1

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) = {3/10x(3-x) if 0<=x<=2
.........{0 otherwise
a) Find the standard deviation of X to four decimal
places.
b) Find the mean of X to four decimal places.
c) Let y=x2 find the probability density function
fy of Y.

Let X be the random variable with probability density function
f(x) = 0.5x for 0 ≤ x ≤ 2 and zero otherwise. Find the
mean and standard deviation of the random variable X.

A random variable X takes values between -2 and 4 with
probability density function (pdf)
Sketch a graph of the pdf.
Construct the cumulative density function (cdf).
Using the cdf, find )
Using the pdf, find E(X)
Using the pdf, find the variance of X
Using either the pdf or the cdf, find the median of
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 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...

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=

Let X be a continuous random variable with the probability
density function f(x) = C x, 6 ≤ x ≤ 25, zero otherwise.
a. Find the value of C that would make f(x) a valid probability
density function. Enter a fraction (e.g. 2/5): C =
b. Find the probability P(X > 16). Give your answer to 4
decimal places.
c. Find the mean of the probability distribution of X. Give your
answer to 4 decimal places.
d. Find the median...

suppose x is a continuous random variable with probability
density function f(x)= (x^2)/9 if 0<x<3 0 otherwise
find the mean and variance of x

The random variable X has probability density function:
f(x) =
ke^(−x) 0 ≤ x ≤ ln 2
0 otherwise
Part a: Determine the value of k.
Part b: Find F(x), the cumulative distribution function of X.
Part c: Find E[X].
Part d: Find the variance and standard deviation of X.
All work must be shown for this question. R-Studio should not be
used.

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