Contents of this Probability theory episode:

Random variable, Continuous random variable, Exponential distribution, Normal distribution, Uniform distribution, Probability, Average, Density function, Distribution function, Expected value, Standard deviation.

Continuous random variables usually measure time, distance and stuff like how many pounds, how many gallons, etc.

Given their nature, there is no point in asking what probability is, because all such probabilities are zero.

This can be proven easily, for instance by visiting a pub where they serve beer on tap. We will get either more or – more likely – less, but for sure, never the exact amount of beer that supposed to be in the glass.

Well, this is not the most precise way to make this point, but perhaps it helps us remember that in the case of continuous random variables, only intervals have any place in questions like or or

We can get the exact probabilities by using the distribution function or the density function, and in most cases we can decide which one to use. If you feel an irresistible desire towards computing integrals, go ahead and use the density function. However, your degree of suffering will be less, if you opt for the distribution function.

In Step 1 we convert the probability to a distribution function and in Step 2 we find the actual distribution function.

The distribution of events over time or distance.

UNIFORM DISTRIBUTION (ez csak úgy van)

Private Ryan is expecting a phone call between 2pm and 7pm, in each moment the chance of being called is equal. What is the probability of getting the call before 4pm?

=what time it is

a=10 b=15

The distribution function of the uniform distribution is:

here a=10 and b=15

Call received before noon:

[K1]

EXPONENTIAL DISTRIBUTION

Usually 12 customers enter the bank every hour. What is the probability of 10 minutes passing without anyone entering?

=elapsed time, minutes

0 10 minutes

If nobody enters for 10 minutes, that means the elapsed time between two customers is longer than 10 minutes.

, so we want to calculate this probability: .

The expected number of customers entering the bank in an hour is 12, so the time between customers is 0/12=5 minutes, and the expected value is

minutes, and therefore [K2]

The distribution function of the exponential distribution:

[K3] here [K4] [K5]

Nobody enters for 10 minutes:

[K6]

NORMAL DISTRIBUTION (the distribution of quantities)

The number of customers entering the bank in a day follows the normal distribution, with an expected value of 560 customers, and a standard deviation of 40.

This means, that usually 560 customers visit this bank daily. Sometimes more, sometimes fewer.

However, it is rare to have far more or far fewer customers in a day.

The density function of the normal distribution is:

This is a fantastic function, but unfortunately, it has a little problem. We can't find the integral of this function. I don't mean today, but at all. But no sweat, we haven’t been using the density function for calculating probabilities anyway, instead, we used the distribution function.

Only there is a little trouble. Namely, there is no presentable distribution function.

We can eliminate this minor inconvenience if we introduce a special normal distribution with an expected value of zero and a standard deviation of one.

This is called the standard normal distribution. Its density function is:

and its distribution function is defined as a table, represented by the symbol of .

Let's see the table.

Well, we have two tables here. But no need to panic, the two tables are practically the same, as we will see shortly.

The density function of the standard normal distribution is the so called Gaussian curve or bell curve.

The first table contains the values of the distribution function, in other words, the size of the area under the curve from negative infinity to z.

If z=0, then this is exactly half of the total area.

Since the area under the curve of density functions is 1, half of it is 0.5.

If z is slightly bigger than 0, then the area is slightly bigger, too.

Here comes the other table. The only difference here is that the areas start from 0.

Therefore, the areas are exactly 0.5 less than in the first table.

It doesn't matter which table we use for solving the problem, but if there is a choice, it may be worth picking the first one.

Finally, there is one more thing here.

is indicated in the figure by this area.

is this area.

We can see that these two areas complement each other: they add up to the total area.

.

Well, this is splendid, but now let's continue solving the problem.

The normal distribution can be converted into the standard normal distribution by subtracting its expected value from , and divide it by the standard deviation. Based on this, the distribution function of the normal distribution is:

Here we have a normal distribution with an expected value of 560, and a standard deviation of 40.

The probability of having less than 616 customers on a given day is:

[K7]

.

If we use the first table, we will find the probability we are looking for.

If we use the second table, we have to add 0.5 to the value found there.

Let’s see another one of these.

The probability of having less than 480 customers on a given day is:

[K8]

Probability theory episode