Question

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?

Answer #1

1)

For any continuous distribution, probability of random variable X equal to any particular value is zero.

For example in Normal distribution, P(X=0) = 0

As probability of (X<0) is area under curve between 0 and - infinity.

2) Yes

In bernoulli distribution we have two outcomes 'success' and 'failure'.

In an experiment of rolling a fair die, it is independent Bernoulli trial. Out of six outcomes, for example when 1 occurs then it's a 'success' and otherwise it's a 'failure' for remaining 5 outcomes.

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 ).

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

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).

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.

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 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.

Suppose that X is a continuous random variable with a
probability density function that is a positive constant on the
interval [8,20], and is 0 otherwise.
a. What is the positive constant mentioned
above?
b. Calculate P(10?X?15).
c. Find an expression for the CDF FX(x).
Calculate the following values.
FX(7)=
FX(11)=
FX(30)=

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 .

Mean, Variance and Standard Deviation of a Continuous Random
Variable
37. Consider the density function (?)=3? 2 on the interval
[0,1]. Find the expected value E(X), the variance Var(X) and the
standard deviation σ(X) for the density function and round your
answers to four decimal places [Clearly state the method you used
and how you calculated your result if you used the calculator]
38.Find the median of the random variable with the probability
density function given in question 37 round...

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