example problems on probability density function: Say we have a continuous random variable whose probability density function is given by f(x) = x + 2, when 0 < x ≤ 2. We want to find P(0.5 < X < 1). Then we integrate x + 2 within the limits 0.5 and 1. This gives us 1.375. Thus, the probability that the continuous random variable lies between 0.5 and 1 is 1.375.
cumulative distribution function: You invite people at 8pm. You expect 100 people. Let's assume that people's arrival is gaussianly distributed (which is not) with a mean of 10pm and a standard deviation of 1 hour. The cumulative distribution tells you that by 11pm you will have ~84 people and by 12pm ~98 people.
mutually exclusive events:
Some of the examples of the mutually exclusive events are:
- When tossing a coin, the event of getting head and tail are mutually exclusive. Because the probability of getting head and tail simultaneously is 0.
- In a six-sided die, the events “2” and “5” are mutually exclusive. We cannot get both the events 2 and 5 at the same time when we threw one die.
- In a deck of 52 cards, drawing a red card and drawing a club are mutually exclusive events because all the clubs are black.
