Basically, the variance is the expected value of the squared difference between each value and the mean of the distribution. In both cases, we’re “summing” over all possible values of the random variable and multiplying each squared difference by the probability or probability density of the value.

To find the interquartile range (IQR), ​first find the median (middle value) of the lower and upper half of the data. These values are quartile 1 (Q1) and quartile 3 (Q3). The IQR is the difference between Q3 and Q1.

The shape of a distribution is described by its number of peaks and by its possession of symmetry, its tendency to skew, or its uniformity. (Distributions that are skewed have more points plotted on one side of the graph than on the other.) PEAKS: Graphs often display peaks, or local maximums.

For example, a flat distribution can be said either to have no tails, or to have short tails. A normal distribution is usually regarded as having short tails, while an exponential distribution has exponential tails and a Pareto distribution has long tails.

=dFX(x)dx=F′X(x),if FX(x) is differentiable at x. is called the probability density function (PDF) of X. Note that the CDF is not differentiable at points a and b….Solution

1. To find c, we can use Property 2 above, in particular.
2. To find the CDF of X, we use FX(x)=∫x−∞fX(u)du, so for x<0, we obtain FX(x)=0.

The expected value (EV) is an anticipated value for an investment at some point in the future. In statistics and probability analysis, the expected value is calculated by multiplying each of the possible outcomes by the likelihood each outcome will occur and then summing all of those values.

The sum of the probabilities in a probability distribution is always 1. A probability distribution is a collection of probabilities that defines the likelihood of observing all of the various outcomes of an event or experiment.

A standard normal distribution is a normal distribution with zero mean ( ) and unit variance ( ), given by the probability density function and distribution function. (1) (2) over the domain .

The expected value of a random variable is the weighted average of all possible values of the variable. The weight here means the probability of the random variable taking a specific value. What is the expected value of the length of a carrot? The random variable here is the length of a carrot.