Question

Consider a continuous random variable X with the probability density function f X ( x ) = |x|/C , – 2 ≤ x ≤ 1, zero elsewhere. a) Find the value of C that makes f X ( x ) a valid probability density function. b) Find the cumulative distribution function of X, F X ( x ).

Answer #1

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...

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 .

6. A continuous random variable X has probability density
function
f(x) =
0 if x< 0
x/4 if 0 < or = x< 2
1/2 if 2 < or = x< 3
0 if x> or = 3
(a) Find P(X<1)
(b) Find P(X<2.5)
(c) Find the cumulative distribution function F(x) = P(X< or
= x). Be sure to define the function for all real numbers x. (Hint:
The cdf will involve four pieces, depending on an interval/range
for x....

1 (a) Let f(x) be the probability density function of a
continuous random variable X defined by
f(x) = b(1 - x2), -1 < x < 1,
for some constant b. Determine the value of b.
1 (b) Find the distribution function F(x) of X . Enter the value
of F(0.5) as the answer to this question.

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 ).

If f(x) is a probability density function of a continuous random
variable, then f(x)=?
a-0
b-undefined
c-infinity
d-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.

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]

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.

1. Consider a continuous random variable X with a probability
density function that is normal with mean 0 and standard deviation
What is the probability that X = 0? Explain your answer.
2. Is each outcome of the roll of a fair die an independent
Bernoulli trial? Why or why not?

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