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.
1. A continuous random variable X is defined by
(3+x)
f(x)
- 3 sxs- 1
%3D
16
(6-2r)
-1sxs1
16
(3-x)
-1sxs3
16
a. Verify that f(x) is density. Explain
b. Find the mean
II
Let Xi- NcO, ) andn=2 aud
T=X,+ Xz Show that Tis a
%3D
Sulfficiesry
Statistic.
1)Let x and y be two continuous random variables whose function is the probability density joint is given by : a)Draw the relationship between the variables x and y on the Cartesian axes .b)Calculate the marginal pdfs px(X)and py(Y).c)Are the v.a.s x and y independent ?
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