Question

1 (a) Let f(x) be the probability density function of a continuous random variable X defined by

f(x) = b(1 - x^{2}), -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.

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...

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.

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 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 ).

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 ).

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 .

Let X be a continuous random variable with a probability density function
fX (x) = 2xI (0,1) (x) and let it be the function´
Y (x) = e^−x
a. Find the expression for the probability density function fY (y).
b. Find the domain of the probability density function fY (y).

A continuous random variable X has the following
probability density function F(x) = cx^3, 0<x<2 and 0
otherwise
(a) Find the value c such that f(x) is indeed
a density function.
(b) Write out the cumulative distribution function of
X.
(c) P(1 < X < 3) =?
(d) Write out the mean and variance of X.
(e) Let Y be another continuous random variable such
that when 0 < X < 2, and 0 otherwise. Calculate
the mean of Y.

Suppose that X1 and X2 are independent continuous random
variables with the same probability density function as: f(x) = ( x
2 0 < x < 2, 0 otherwise. Let a new random variable be Y =
min(X1, X2,).
a) Use distribution function method to find the probability
density function of Y, fY (y).
b) Compute P(Y > 1).

A random variable X has probability density function f(x)
defined by f(x) = cx−6 if x > 1, and f(x) = 0, otherwise.
a. Find the constant c.
b. Calculate E(X) and Var(X).
c. Now assume Z1, Z2, Z3, Z4 are independent RVs whose
distribution is identical to that of X. Compute E[(Z1 +Z2 +Z3
+Z4)/4] and Var[(Z1 +Z2 +Z3 +Z4)/4].
d. Let Y = 1/X, using the formula to find the pdf of Y.

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