Consider the beta distribution with parameters ( a , b ) . Show that a. when a > 1 and b > 1 , the density is unimodal (that is, it has a unique mode ) with mode equal to ( a − 1 ) ( a + b − 2 ) ; b. when a ≤ 1 , a ≤ 1 , and a + b < 2 , the density is either unimodal with mode at 0 or I or U-shaped with modes at both 0 and 1; c. when a = 1 = b . all points in [ 0 , 1 ] are modes.
Consider the beta distribution with parameters ( a , b ) . Show that a. when a > 1 and b > 1 , the density is unimodal (that is, it has a unique mode ) with mode equal to ( a − 1 ) ( a + b − 2 ) ; b. when a ≤ 1 , a ≤ 1 , and a + b < 2 , the density is either unimodal with mode at 0 or I or U-shaped with modes at both 0 and 1; c. when a = 1 = b . all points in [ 0 , 1 ] are modes.
Solution Summary: The author explains that the density is unimodal when a>1 and b>1.
Consider the beta distribution with parameters
(
a
,
b
)
. Show that
a. when
a
>
1
and
b
>
1
, the density is unimodal (that is, it has a unique mode) with mode equal to
(
a
−
1
)
(
a
+
b
−
2
)
;
b. when
a
≤
1
,
a
≤
1
, and
a
+
b
<
2
, the density is either unimodal with mode at 0 or I or U-shaped with modes at both 0 and 1;
c. when
a
=
1
=
b
. all points in
[
0
,
1
]
are modes.
Definition Definition Probability of occurrence of a continuous random variable within a specified range. When the value of a random variable, Y, is evaluated at a point Y=y, then the probability distribution function gives the probability that Y will take a value less than or equal to y. The probability distribution function formula for random Variable Y following the normal distribution is: F(y) = P (Y ≤ y) The value of probability distribution function for random variable lies between 0 and 1.
Find the critical value of t for a t-distribution with 30 degrees of freedom such that the area between −t and t is 99%.
Suppose that the random variables X and Y have a joint density function given by:
f(x,y) = {c(2x+y) for 2≤x≤6 and 0≤y≤5, 0 otherwise
P(3 < X < 5, Y >1),
P(X < 3),
P(X +Y > 5),
Find the joint distribution function (cdf),
Suppose that two random variables X and Y have the joint PDFfXY(u, v) = {60(u^2)v u ≥ 0, v ≥ 0, and u + v ≤ 1, 0 otherwise.(a) Are X and Y independent?(b) What is the marginal distribution fX(t)?(c) What is Pr[X ≥ Y]? (To set up the right integral, it might help you to draw the rangeof (X, Y) in the uv-plane and identify the region within that range where u ≥ v.)
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