Uniform Distribution and Probability

Uniform Distribution:
A uniform distribution is a probability distribution where each value in the range is equally likely to be drawn. The probability density function of a continuous uniform distribution is defined as:

f(x) = 1/(b-a) for a ≤ x ≤ b

where a and b are the two endpoints of the range of the distribution.

Random Variable:
A random variable is a variable whose value is determined by the outcome of a random event. In other words, it is a function that assigns a numerical value to each possible outcome of a random experiment.

Equal Probability:
When a random variable assumes n values with equal probability, it means that each value has the same chance of being selected. In other words, the probability of each value is 1/n.

For example, if we have a random variable X that represents the outcome of rolling a fair six-sided die, then X can take on the values 1, 2, 3, 4, 5, or 6 with equal probability. That is, the probability of rolling any particular value is 1/6.

Unequal Probability:
If a random variable assumes n values with unequal probability, it means that some values are more likely to be selected than others. In this case, the sum of the probabilities of all the values must still equal 1.

For example, suppose we have a random variable Y that represents the outcome of flipping a coin twice. Y can take on the values HH, HT, TH, or TT. However, each value does not have an equal probability. The probabilities of each value are:

P(HH) = 1/4
P(HT) = 1/4
P(TH) = 1/4
P(TT) = 1/4

Conclusion:
Therefore, in the case of a uniform distribution where the random variable assumes n values, each value has an equal probability of being selected.