Summary
- If X is a discrete random variable, the function given by f (x) = P(X = x)
- The sum of all the probability will always equal to one.
- The cumulative distribution function, F(x) for a random variable X is defined as the probability of certain events that exist.
- Cumulative probabilities provide, for each value x, the probability of a result less than or equal to X, .
Discrete random variable are random variable that can take on distinct and separate variable.
A random variable is a function defined on the sample space. However, a Discrete random variables is a variable that can only take a finite or countable number of values, and have a positive probability of taking each one.
If X is a discrete random variable, the function given by:
f (x) = P(X = x)
For each value of x within the range of X is called the probability distribution (or probability function) of X.
A probability distribution is said to be a table of probabilities that possible outcomes of an experiment. It will look something like this.
x | -2 | 2 | 1 | 3.5 |
p(X) | 0.21 | 0.34 | 0.24 | 0.21 |
The sum of all the probabilities will always equal to one.
Eg. adding the probabilities above shows this:
0.21 + 0.34 + 0.24 + 0.21 = 1
Probability Density Function
The cumulative distribution function, F(x) for a random variable X is defined as the probability of certain events that exist.
Example #1
Q. A discrete random variable X has the following probability distribution:
x | -1 | 0 | 1 | 4 |
p(X) | 0.2 | 0.5 | a | 0.1 |
Compute each of the following quantities:
1) a
2) P(0)
3) P(X > 0)
4) P(X ≥ 0)
5) P(X ≤ −2)
Solution:
1) In order to find a we will add the values and put them equal to 1
0.2 + 0.5 + a +0.1 = 1
a = 0.2
2) Next, we will find P(0) we will look directly at the table and p(0) has 0.5 probability. Thus the answer will be 0.5.
P(0) = 0.5
3) For p(X > 0), since its only greater than 0 we won’t include 0, but will add the probabilities of P(X = 1) + P(X = 4). So, we will add ”a” which we calculated above to be 0.2 and 0.1 to get
0.2 + 0.1 = 0.3
P(X > 0) = 0.3
4) For P(X ≥ 0) while calculating this, we will include 0 as it says greater than equal to 0. Thus, we get:
P(X ≥ 0) = P(X = 0) + P(X = 1) + P(X = 4)
= 0.5 + 0.2 + 0.1
= 0.8
5) For P(X ≤ −2), since there is no value as -2 thus the answer will be 0.
Cumulative Distribution Function
The Cumulative Distribution Function (also written as CDF) is the cumulation of the probability of all the outcomes upto a given value. Assume we have a random variable X. Cumulative probabilities provide, for each value x, the probability of a result less than or equal to X. This can be represented as:
Example #2
x | 1 | 2 | 3 | 4 |
f(x) | 0.1 | 0.2 | 0.3 | 0.4 |
1) P( X ≤ 1 )
2) P( X ≤ 2 )
3) P( X ≤ 3 )
4) P( X < 4 )
Solution:
1) In order P( X ≤ 1 ), we can see that there isn’t a value lower than 1, so we will just take the value of 1 to get 0.1.
2) P( X ≤ 2 ), since it is less than equal to 2. We will take the value of 1 and 2.
P( X ≤ 2 ) = P(X = 1)+ P(X = 2)
= 0.1 + 0.2
= 0.3
3) P( X ≤ 3 ) = P( X ≤ 1 ) + P( X ≤ 2 ) + P( X ≤ 3 )
= 0.1 + 0.2 + 0.4
= 0.7
4) P( X <4 ) = P( X ≤ 1 ) + P( X ≤ 2 ) + P( X ≤ 3 ) (since, it’s not less than equal to thus, we will just take probabilities till 3)
=0.1 + 0.2 + 0.4
=0.7