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Cumulative Probability

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Cumulative Probability refers to the probability that a certain event occurs less than or equal to a certain number of times. It is obtained by adding up the probabilities of the random variable taking values less than or equal to the specified number.

Example:

Suppose that two coins are tossed. Here the possible events are {HH, HT, TH, TT}. Let X denote the number of heads obtained.

The probability of X=1, that is getting one heads is 2/4 = 0.5 since we have two favorable outcomes (HT, TH) out of four. So we obtain the probability distribution table as follows,

X = number of HeadsProbability P(X=x)
01/4 = 0.25
12/4 = 0.5
21/4 = 0.25

We can obtain the cumulative probability for a particular value of X by adding up the probabilities of the current and previous values of X.

For example, P(X\leq1) = P(X=0) + P(X=1) = 0.25 + 0.5 = 0.75

In a similar manner we obtain the cumulative probability distribution as follows:

X = number of HeadsProbability P(X=x)Cumulative Probability P(X\leqx)
01/4 = 0.250.25
12/4 = 0.50.25+0.5 = 0.75
21/4 = 0.250.25+0.5+0.25 = 1

How do you calculate the cumulative probabilities for a continuous distribution?

As seen above we can calculate the cumulative probabilities for a discrete random variable by adding up all the previous probabilities.

\text{Cumulative Probability = }\sum_{-\infty}^{x} P(X=x)

For a continuous random variable, given the probability distribution function (pdf) f(x) we can obtain the cumulative probability distribution function F(x) by integrating the pdf from minus infinity to X.

\text{Cumulative Probability = } P(X\leq x) = \int_{-\infty}^{x} f(x)dx

Example: Suppose that a continuous random variable X has the pdf,

f(x) = 3x^2 \text{ where } 0\leq x \leq 1

Find the cumulative probability distribution function.

Solution: F(x) = P(X\leq x) = \int_{0}^{x} f(x)dx = [x^3]_0^x = x^3 which is the required cumulative probability distribution function.

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