# What is the difference between a normal profile of a random variable and normal pdf of a random variable. What is the median value of a random variable having a normal pdf.

What is the difference between a normal profile of a random variable and normal pdf of a random variable. What is the median value of a random variable having a normal pdf.

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From the given information,
These properties are demonstrated by several factors, such as weight, shoe sizes, foot lengths, and other human physical characteristics. The symmetry implies that the variable is just as likely to take a value below its mean from a certain distance as it is to take a value above its mean from the same distance. The shape of the bell suggests that values similar to the mean are more probable, and values far from the mean are increasingly unlikely to be taken in any direction
A random variable is a function X with a particular domain and range that takes all possible outcomes from the function of the event or distribution. The random variable may be continuous or discrete in nature and any distribution function may or may not be followed. To explain these bell-shaped data distributions, researcher use a mathematical model with a smooth bell-shape curve.
These models are called regular curves or distributions that are normal. In several different types of common measurements, they were first named "normal" because the pattern occurred. This is a normal rv profile.
The normal distribution is by far the most relevant distribution of probabilities. The Central Limit Theorem (CLT) that we will discuss later in the book is one of the key explanations for that. To give you an idea, the CLT states that the distribution of the sum would be roughly normal under some conditions if you add a large number of random variables.
The meaning of this outcome is that many random variables can be expressed as the sum of a large number of rankings in real life.
Consider a random variable X. with expected value as E(X) and variance as $$\displaystyle{V}{a}{r}{\left({X}\right)}\frac{{{x}-{E}{\left({X}\right)}}}{{{V}{\left({X}\right)}}}$$ then is considered as normalization or normal profile of a random variable.
Example:
Random Variable X is said to follow Binomial distribution. $$\displaystyle{X}~{B}\in{\left({n},{p}\right)}$$ and it is known that expected value of binomial distribution is $$\displaystyle{E}{\left({X}\right)}={n}{p}$$ and Variance, $$\displaystyle{V}{a}{r}{\left({X}\right)}={n}{p}{q}$$. Then random variable X in the form $$\displaystyle\frac{{{X}-{n}{p}}}{{{n}{p}{q}}}$$ has a normal profile
A random variable X is said to follow Normal distribution then X having probability distribution function (pdf)
$$\displaystyle f{{\left({X}={x}\right)}}=\frac{1}{{\sigma\sqrt{{{2}\pi}}}}\frac{ \exp{{\left(-{1}\right)}}}{{2}}{\left(\frac{{{x}-\mu}}{{\sigma}}\right)}^{2},-\infty<{x}<\infty,-\infty<\mu<\infty,\sigma>{0}$$
is known as normal pdf of a random variable. where $$\mu$$ is mean and $$\sigma$$ is standard deviation of the normal distribution