What does expectation mean in probability?

The expectation or expected value of a random variable is a single number that tells you a lot about the behavior of the variable. Roughly, the expectation is the average value of the random variable where each value is weighted according to its probability.

What you know about the mathematical expectation of a random variable write down its properties in detail?

Mathematical expectation, also known as the expected value, which is the summation of all possible values from a random variable. It is also known as the product of the probability of an event occurring, denoted by P(x), and the value corresponding with the actually observed occurrence of the event.

What is linearity of expectation?

Linearity of expectation is the property that the expected value of the sum of random variables is equal to the sum of their individual expected values, regardless of whether they are independent.

Why is it important to know the expected value of a probability distribution?

An expected value gives a quick insight into the behavior of a random variable without knowing if it is discrete or continuous. Therefore, two random variables with the same expected value can have different probability distributions.

What is the expected value of a normal distribution?

The expected value µ = E(X) is a measure of location or central tendency. The standard deviation σ is a measure of the spread or scale. The variance σ2 = Var(X) is the square of the standard deviation.

Why random variable is a function?

A (real-valued) random variable, often denoted by X (or some other capital letter), is a function mapping a probability space (S, P) into the real line R. (The set of possible values of X(s) is usually a proper subset of the real line; i.e., not all real numbers need occur.

What is the formula for finding the mean of a discrete random variable?

The mean μ of a discrete random variable X is a number that indicates the average value of X over numerous trials of the experiment. It is computed using the formula μ=Σx P(x).

Why do we need mathematical expectation?

The mathematical expectation of an indicator variable can be zero if there is no occurrence of an event A, and the mathematical expectation of an indicator variable can be one if there is an occurrence of an event A. Thus, it is a useful tool to find the probability of event A.

What do you understand by expectation mathematical expectation expected value of a random variable?

The expected value, or mathematical expectation E(X) of a random variable X is the long-run average value of X that would emerge after a very large number of observations. We often denote the expected value as µX, or µ if there is no confusion.

Why is expectation a linear operator?

The expectation operator has inherits its properties from those of summation and integral. In particular, the following theorem shows that expectation preserves the inequality and is a linear operator. Theorem 1 (Expectation) Let X and Y be random variables with finite expectations.

How do you find the normal distribution in statistics?

Note that the normal distribution is actually a family of distributions, since µ and σ determine the shape of the distribution. πσ µσ • The notation N(µ, σ2) means normally distributed with mean µ and variance σ2. If we say X ∼ N(µ, σ2) we mean that X is distributed N(µ, σ2).

How do you find the probability of two independent events?

Theorem 1 : If A and B are two independent events associated with a random experiment, then P (A⋂B) = P (A) P (B) Probability of simultaneous occurrence of two independent events is equal to the product of their probabilities.

What is the first known property of the normal distribution?

The First Known Property of the Normal Distribution says that: given random and independent samples of N observations each (taken from a normal distribution), the distribution of sample means is normal and unbiased (i.e., centered on the mean of the population), regardless of the size of N.

How do you find the product of probability and probability?

P (AB) = P (A) * P (B) Theorem 1 : If A and B are two independent events associated with a random experiment, then P (A⋂B) = P (A) P (B) Probability of simultaneous occurrence of two independent events is equal to the product of their probabilities.