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.
Patterns in Floor Tiling A square floor is to be tiled with square tiles as shown. There are blue tiles on the main diagonals and red tiles everywhere else.
In all cases, both blue and red tiles must be used. and the two diagonals must have a common blue tile at the center of the floor.
If 81 blue tiles will be used, how many red tiles will be needed?
For what numbers in place of 81 would this problem still be solvable?
Find an expression in k giving the number of red tiles required in general.
At a BBQ, you can choose to eat a burger, hotdog or pizza. you can choose to drink water, juice or pop. If you choose your meal at random, what is the probability that you will choose juice and a hot dog? What is the probability that you will not choose a burger and choose either water or pop?
a card is drawn from a standard deck of 52 cards. If a card is choosen at random, what is the probability that the card is a)heart b)a face card or c)a spade or 10
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