- 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.
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.
Q. A discrete random variable X has the following probability distribution:
Compute each of the following quantities:
3) P(X > 0)
4) P(X ≥ 0)
5) P(X ≤ −2)
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
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:
1) P( X ≤ 1 )
2) P( X ≤ 2 )
3) P( X ≤ 3 )
4) P( X < 4 )
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
3) P( X ≤ 3 ) = P( X ≤ 1 ) + P( X ≤ 2 ) + P( X ≤ 3 )
= 0.1 + 0.2 + 0.4
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