5. A continuous random variable X is uniformly distributed over
(0,10). Compute the probability that an...
5. A continuous random variable X is uniformly distributed over
(0,10). Compute the probability that an observed value of X will be
within one standard deviation of its mean.
Let X and Y be independent continuous random variables, with
each one uniformly distributed in the interval from 0 to1. Compute
the probability of the following event.
XY<=1/7
The probability density function of the continuous random
variable X is given by
fX (x) = kx, (0 <= x <2)
= k (4-x), (2 <= x <4)
= 0, (otherwise)
1) Find the value of k
2)Find the mean of m
3)Find the Dispersion σ²
4)Find the value of Cumulative distribution function FX(x)
Consider a continuous random variable X with the probability
density function f X ( x ) = x/C , 3 ≤ x ≤ 7, zero elsewhere.
Consider Y = g( X ) = 100/(x^2+1). Use cdf approach to find the cdf
of Y, FY(y). Hint: F Y ( y ) = P( Y <y ) = P( g( X ) <y )
=
Let x be a continuous random variable
that is normally distributed with a mean of 65 and a standard
deviation of 15. Find the probability that
x assumes a value:
less than 48
greater than 87
between 56 and 70
Let X and Y be uniformly distributed independent random
variables on [0, 1].
a) Compute the expected value E(XY ).
b) What is the probability density function fZ(z) of Z = XY
?
Hint: First compute the cumulative distribution function FZ(z) =
P(Z ≤ z) using a double integral, and then differentiate in z.
c) Use your answer to b) to compute E(Z). Compare it with your
answer to a).
1)Using R, construct one plot of the density function for a
uniformly distributed continuous random variable defined between 0
and 6.
Make sure you include a main title, label both axis correctly
and include a legend in your figure.
The main title needs to include your last name, also include
your R code with your answer.
2) Using R, construct one plot of the density function for a
Chi-square distributed random variable.
The plots should contain six lines corresponding to...
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).