Page 142 - Data Science class 10
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Roll 1 2 3 4 5 6
Odds 1/6 1/6 1/6 1/6 1/6 1/6
Intuitively, we would expect the sum of a single die to be the average of the possible outcomes, i.e.
(1+ 2 + 3 + 4 + 5 + 6)
S = = 3.5
6
And so we would predict the sum of a two die to be twice that of one die, i.e., we would predict the expected value
to be 7. If we consider the possible outcomes from the throw of two dice:
Outcome of First Die
1 2 3 4 5 6
Outcome of Second Die 2 3 4 5 6 7 10
6
1
4
5
3
7
2
8
7
6
4
9
8
3
5
5
4
6
8
9
7
9
5
8
12
11
9
7
6 6 7 8 10 10 11
And so if we define X as a random variable denoting the sum of the two dices, then we get the following distribution:
X 2 3 4 5 6 7 8 9 10 11 12
P(X=X) 1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36
So then we compute the expected value, using the following formula:
E(X) = ∑x(P(X)=x)
2 × 1 3 × 2 4 × 3 5 × 4 6 × 5 7 × 6 8 × 5 9 × 4 10 × 3 11 × 2 11 × 1
= + + + + + + + + + +
36 36 36 36 36 36 36 36 36 36 36
(2 + 6 + 12 + 20 + 30 + 42 + 40 + 36 + 30 + 22 + 12)
=
36
252
= = 7
36
The density functions are mathematical functions that describe the probability distribution of a random variable X.
Probability Mass Functions (PMF) describe the probability of a random variable X taking on a particular value x, and
it is only applicable for discrete distributions.
Discrete probability distribution is of three types:
• Bernoulli Distribution
• Binomial Distribution
• Poisson Distribution
Let us discuss these in detail.
140 Touchpad Data Science-X

