Khan Open Course-Statistical Study Notes: (3) random variables, probability density, two-item distribution, and expected values

Random Variable

A random variable is different from a common data variable. It is usually expressed in uppercase letters, such as X, Y, and Z. It is not a parameter but a function, that is, a function. For example, the random variable X indicating whether it is raining tomorrow is shown below. For example, if X = is a vehicle passing through an intersection every hour, a random variable is a description rather than a variable in the equation.

There are two types of random variables: discrete and continue ). For example, the preceding example can be used for enumeration, while the continuous random variable value is infinite.

Probability Density Function

Probability. Taking roll dice as an example, P (X = 6) = 1/6, P (X> = 5) = 1/3, that is, the probability of a 6-point dice is 1/6, the probability of a dice that is greater than or equal to 5 is 1/3. This is an example of discrete probability.

For continuous, such as tomorrow's rainfall. The probability density function is used as an example of distribution.

What is P (X = 2)? 0.5? No. Precise rainfall: 2.00000 ......, The probability is 0. For continuous random variables, probability statistics are a range, such as P (| X-2 | <0.3), equivalent to calculating area. If f (x) represents a random variable

Binary Distribution

Binary distribution: binomial distribution. A more familiar name is normal distribution. Random Variables are in two states: Front or back of a coin, shot or miss. If it is fair and random, for example, throwing a coin, the probability of each status appears is 0.5. For shooting, it may be P (shoot) = 0.7, P (miss) = 0.3.

How to calculate P (X = n) and n is the number of times a certain State occurs. Assuming that a total of N shots are shot (N = 6), how many combinations are possible, such as a combination of two hits. Simply put, we have two letters, A and B, and six spaces. How many combinations can we use. It is 6 × 5. If there are letters A, B, and C, there are 6 × 5 × 4, that is, N! /(N-n )!

In the probability calculation, the order of A and B does not affect, that is, there is no order, but it is also divided by n! (A, B, or A, B, and C are arranged and combined:

We get the number of combinations. What is the probability of each combination? In 6, 2 is P (shoot) p (miss) P (miss ), we can multiply the probability of occurrence at each position, that is, p ^ n × (1-p) ^ (N-n). The total probability is:

In fact, you don't need to memorize it. As long as you know the computing principle, it is easy to deduce it.

These probabilities are suitable for computing and drawing in Excel. I have never had a trick in Excel. If I select a certain unit and then use F4, the position will not be moved when I copy the formula.

Expected Value E (X)

Expected value Exptected value of a random varable is actually the population mean. In some cases, it is always infinite. For example, the result of countless times of coin placement can be obtained through the sum of frequency X values.

E (X) of binary Distribution)

If it is a binary distribution, n indicates the number of times, E (X) = np. This derivation process is very interesting.

Variance of Two-item distribution)

Similar to expectations, it belongs to the mind gymnastics, and its basic approach is similar. The variance is np (1-p ). This part is not the Khan open class. It is about the calculation formula of the binary variance involved in the normal distribution. Let's have a look.

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