Questions
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Recognize that
As you stated, for a continuous random variable
The moment generating function is
(a) Therefore,
(b) The probability is
(c) In the general case, the mean is
The mean is also the "first moment"
The second moment is
The variance
See below.
The parameters
The
The moment-generating function
If
#(d^km(t))/(dt^k)]_(t=0)=m^(k)(0)=mu_k^'#
In other words, if you find the
Then we find that various probability distributions have their own unique moment-generating function.
Let's first see the number of ways we can pick 4 cards from a pack of 52:
How many ways can we draw 4 cards and have exactly 2 of them be spades? We can find that by choosing 2 from the population of 13 spades, then choosing 2 cards from the remaining 39 cards:
This means the probability of drawing exactly 2 spades on a 4 card draw from a standard deck is:
Please see below and refer to the diagram included.
Problem:
In a class of In a class of 50 students, 18 take choir, 26 take concert band, and 2 students take both. How many students are not in either choir or concert band?, 18 take choir, 26 take concert band, and 2 students take both.
How many students are not in either choir or concert band?
Summary:
The Universe is the entire collection of students in the class under consideration:
Universe: 50 students in a class.
Concert Band: cb = 26 students.
Choir: c = 18 students.
The drawing below is an illustration of a Venn Diagram. Mr. Venn is given credit for the invention of a way to deal with problems like this one.
The rectangle contains the "universe", the total collection of people or items under consideration.
Each circle contains numbers that represent the people or items belonging to a particular subgroup. There are two subgroups:
a circle labelled cb contains the total number of people in the universe who are taking concert band.
a circle labelled c contains the total number of people in the universe who are taking choir.
the overlapping area of the two circles contains the number of people who are taking both subjects c, and cb.
Adding the numbers in each part of the circles in the diagram gives the number of people who are in either one of the two classes or both of them.
number in cb + number in c + number in both + number not in either one of the two classes makes-up the total universe, 50.
People taking the specified classes:
People not taking either of the specified classes:
Summary:
Let,
and a Black-coloured, resp.
Then, the Reqd. Prob. is,
We know that,
For P(A), there are
selected in
There are
Similarly,
Note that,
an ace and black-coloured, i.e., a black-coloured ace.
Out of
Clearly,
From
derived by Respected Andrea S. !
Enjoy Maths.!
a. 36,036 ways
b. 278,460 ways
There are 7 boys and 13 girls in the class.
When picking teams, we're looking at combinations (we don't care in what order the players are picked). The general formula is:
a
We want a team with 2 boys (from a population of 7) and 6 girls (from a population of 13):
b
We want a team of 6 with at least 1 boy and 1 girl. So let's first have 1 guaranteed boy and 1 guaranteed girl:
Now we need to fill in the remaining team. We can pick any of the remaining 18 students to fill in the remaining 4 spots:
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