Statistics basic question: Probability distrbution

In summary, we are given a random variable X with three possible values: 0, 2, and 4. Its probability distribution is as follows: the probability of getting a 0 is 0.7, the probability of getting a 2 is 0.1, and the probability of getting a 4 is 0.2. We are also given two independent random variables X1 and X2 with the same distribution. Using this, we can determine the probability distribution for X(bar)2, which is equal to the sum of X1 and X2 divided by 2. From this distribution, we can find the expectation and variance of X(bar)2. To work out the variance, we can use the
  • #1
tamintl
74
0

Homework Statement


A random variable X has three, and only three, possible values, 0; 2; 4
with the following probability distribution:

Probability:
Getting a 0 is 0:7

Getting a 2 is 0:1

Getting a 4 is 0:2

Let X1 and X2 be two independent random variables with this distribution.

(a) Determine the probability distribution for

X(bar)2 = (X1 + X2)/2
and use this distribution to determine the expectation and variance of X(bar)2

(b) Determine the expectation and variance of X and verify that the
variance of X(bar)2 is one-half the variance of X.

Homework Equations


The Attempt at a Solution



X2(bar) = 0 with probability 0.49
X2(bar) = 1 with probability 0.14
X2(bar) = 2 with probability 0.29
X2(bar) = 3 with probability 0.04
X2(bar) = 4 with probability 0.04

not sure how to work out the variance?

Regards as always
Tam
 
Last edited:
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  • #2
tamintl said:

Homework Statement


A random variable X has three, and only three, possible values, 0; 2; 4
with the following probability distribution:

Probability:
Getting a 0 is 0:7

Getting a 2 is 0:1

Getting a 4 is 0:2

Let X1 and X2 be two independent random variables with this distribution.

(a) Determine the probability distribution for

X(bar)2 = (X1 + X2)/2
and use this distribution to determine the expectation and variance of X(bar)2

(b) Determine the expectation and variance of X and verify that the
variance of X(bar)2 is one-half the variance of X.


Homework Equations





The Attempt at a Solution



X2(bar) = 0 with probability 0.49
X2(bar) = 1 with probability 0.14
X2(bar) = 2 with probability 0.29
X2(bar) = 3 with probability 0.04
X2(bar) = 4 with probability 0.04

not sure how to work out the variance?

Regards as always
Tam

You have the probability distribution of [itex] \bar{X}_2.[/itex] Just apply the _definition_ of variance and the formulas for it that you can find in any relevant book, or on-line if you look.

RGV
 

1. What is a probability distribution?

A probability distribution is a mathematical function that describes the likelihood of a specific outcome occurring in a given set of data. It assigns probabilities to all possible outcomes and allows for the calculation of the probability of a specific event or range of events.

2. What are the types of probability distributions?

There are several types of probability distributions, including the normal distribution, binomial distribution, Poisson distribution, and exponential distribution. These distributions differ in their shapes and characteristics, and are used for different types of data and situations.

3. How do I interpret a probability distribution?

A probability distribution can be interpreted by looking at the shape of the curve and the corresponding probabilities for different outcomes. The area under the curve represents the total probability of all possible outcomes, and the height of the curve at a specific point represents the probability of that outcome occurring.

4. What is the difference between a discrete and continuous probability distribution?

A discrete probability distribution deals with data that can only take on certain values, such as whole numbers. A continuous probability distribution deals with data that can take on any value within a range, such as decimals or fractions. The type of distribution used depends on the type of data being analyzed.

5. How are probability distributions used in statistics?

Probability distributions are essential in statistics as they allow for the calculation of probabilities and the prediction of outcomes based on data. They are used to analyze and interpret data, make predictions, and test hypotheses in a variety of fields, including science, finance, and social sciences.

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