Mean and variance calculator for probability distribution

The probability distribution is defined as the possible outcomes for any random event. It is based on the underlying sample space as a set of possible outcomes for any random experiment. And the probability is the measure of uncertainty of phenomena. The random experiment is the result of an experiment, where the outcome can't be predicted.

The two types of probability distribution are listed here.

1. Normal or Cumulative Probability Distribution
2. Binomial or Discrete Probability Distribution

The formulas to find the mean, variance and standard deviation for a distribution are along the lines:

• Mean μ = ∑x · p(x)
• Variance σ² = ∑x² · p(x) - μ²
• Standard Deviation σ = √[∑x² · p(x) - μ²]

Where,

x is the random variable number

p(x) is the probability of the experiment.

Following are the simple steps to calculate the probability distribution mean, variance, and standard deviation.

• Get the number and its probability.
• Make the values in the form of a table to read the data easily.
• Multiply the each value with its probability.
• The sum of resultant products is the mean.
• The sum of each value squared times the probability of the value occurring, minus the mean squared is variance.
• The square root of the variance is the standard deviation.

Example:

Find the mean, standard deviation and variance of the probability distribution?

Solution:

The formula to calculate mean is μ = ∑x · p(x)

= 1(0.2) + 2(0.1) + 3(0.3) + 4(0.05) + 5(0.3) + 6(0.05)

= 0.2 + 0.2 + 0.9 + 0.2 + 1.5 + 0.3

= 3.3

The formul ato find variance of probability distribution is σ² = ∑x² · p(x) - μ²

∑x² · p(x) = (1² . 0.2) + (2² . 0.1) + (3² . 0.3) + (4² . 0.05) + (5² . 0.3) + (6² . 0.0.5)

= 0.2 + 0.4 + 2.7 + 0.8 + 7.5 + 1.8

= 13.4

σ² = 13.4 - 3.3²

= 2.51

Standard deviation σ = √2.51

= 1.584

Check Probabilitycalculatorguru to find the parameters of probability distribution of any data easily & quickly.

1. What does mean by probability?

Probability is defined as the occurrence of an event. It is the ratio of the number of favorable outcomes to the total number of outcomes in an event.

2. How do you calculate probability distribution?

Give each value and its probability in the mentioned input boxes of the calculator and press the calculate button to find the probability distribution mean, standard deviation and variance easily.

3. What are the rules for probability distributions?

The two conditions of the probability for a discrete random variable is function f(x) must be nonnegative for each value of the random variable and second is the sum of probabilities for each value of the random variable must be equal to 1.

4. What are the steps in constructing a probability distribution?

• At first, write the number of widgets in one line.
• Beside each value, write its probability.
• Make a note that, the sum of probabilities should be equal to 1.

For a continuous probability distribution, probability is calculated by taking the area under the graph of the probability density function, written f(x). For the uniform probability distribution, the probability density function is given by f(x)=$\begin{cases} \frac{1}{b-a} \quad \text{for } a \leq x \leq b \\ 0 \qquad \, \text{elsewhere} \end{cases}$. The graph of this function is simply a rectangle, as shown below. Then the area under the graph of f(x) over some interval is also going to be a rectangle, which can easily be calculated as length$\times$width. Furthermore, the expected value and variance for a uniformly distributed random variable are given by E(x)=$\frac{a+b}{2}$ and Var(x) = $\frac{(b-a)^2}{12}$, respectively.

The most important continuous probability distribution is the normal probability distribution. Its graph is bell-shaped and is defined by its mean ($\mu$) and standard deviation ($\sigma$). The mean is the highest point on the curve and the standard deviation determines how flat the curve is. Obviously, this is a much more complicated shape than the uniform probability distribution. The area under it can't be calculated with a simple formula like length$\times$width. So, instead, we rely on the standard normal probability distribution to calculate probabilities for the normal probability distribution.

The standard normal probability distribution (or z distribution) is simply a normal probability distribution with a mean of 0 and a standard deviation of 1. There are three types of probabilities to know how to compute for the z distribution: (1) the probability that z will be less than or equal to a value, (2) the probability that z will be between two values and (3) the probability that z will be greater than or equal to a value. We can find these probabilities using the standard normal table (or z-table), a portion of which is shown below.

Now that we know how to calculate probabilities for the z-distribution, we can calculate probabilities for any normal distribution. We can do this by converting from normal to standard normal, using the formula $z=\frac{x-\mu}{\sigma}$. So, given a problem to calculate probability for a normal distribution, we start by converting the values to z-values. Then, depending on the type of z distribution probability type it is, we rewrite the problem so it's in terms of the probability that z less than or equal to a value. Then we use the z-table to find those probabilities and compute our answer.

The normal probability distribution can be used to approximate probabilities for the binomial probability distribution. This may be necessary in situations where the binomial probabilities are difficult to compute. This calculation is done using the continuity correction factor. This is necessary because the normal distribution is a continuous distribution while the binomial distribution is a discrete distribution. Since the probability of a single value is zero in a continuous distribution, adding and subtracting .5 from the value and finding the probability in between solves this problem.

The exponential probability distribution is useful in describing the time and distance between events. The probability density function for an exponential distribution is given by $ f(x) = \frac{1}{\mu} e^{-x/\mu}$ for x>0. Probabilities for the exponential distribution are not found using the table as in the normal distribution. They involve using a formula, although a more complicated one than used in the uniform distribution. The formula for calculating probabilities in an exponential distribution is $ P(x \leq x_0) = 1 - e^{-x_0/\mu} $.

The t-distribution is similar to the standard normal distribution. They both have a similar bell-shape and finding probabilities involve the use of a table. The main difference is that the t-distribution depends on the degrees of freedom. We have a different t-distribution for each of the degrees of freedom. Another difference is that the t table provides the area in the upper tail whereas the z table provides the area in the lower tail. So use of the t table involves matching the degrees of freedom with the area in the upper tail to get the corresponding t-value.

Continuous probability distributions are probability distributions for continuous random variables. A closely related topic in statistics is discrete probability distributions. Discrete distributions are probability distributions for discrete random variables. Probabilities for discrete probability distributions can be found using the Discrete Distribution Calculator. The most important continuous probability distributions is the normal probability distribution. It is used extensively in statistical inference, such as sampling distributions. Sampling distributions can be solved using the Sampling Distribution Calculator.

How do you find the mean and the variance of the probability distribution?

How do you find the mean and variance of a binomial distribution?