Random Variables and Probability Distributions

Front

What is a random variable?

Back

A random variable is a numerical outcome of a random phenomenon, which can be discrete (taking specific values) or continuous (taking any value within a range).

Front

What is a probability distribution?

Back

A probability distribution describes how the probabilities are distributed over the values of the random variable, indicating the likelihood of each outcome.

Front

What is the shape of a Binomial Distribution?

Back

The shape of a Binomial Distribution can be determined by making a histogram with the discrete random variable on the horizontal axis and the probability on the vertical axis.

Front

What is the formula for the probability of exactly k successes in n trials for a Binomial Distribution?

Back

P(X = k) = (n choose k) * p^k * (1-p)^(n-k), where p is the probability of success.

Front

If X has a mean of 120 and Y has a mean of 100, what is the mean of X - Y?

Back

The mean of X - Y is 120 - 100 = 20.

Front

How does multiplying a random variable by a positive constant 'b' affect its distribution?

Back

The shape is unchanged, but measures of center and spread are both multiplied by 'b'.

Front

What is the mean of the sum of two independent random variables?

Back

The mean of the sum of two independent random variables is the sum of their means.

Front

What is the standard deviation of the sum of two independent random variables?

Back

The standard deviation of the sum of two independent random variables is the square root of the sum of their variances.

Front

What is the variance of a random variable?

Back

The variance is a measure of how much the values of a random variable differ from the mean, calculated as the average of the squared differences from the mean.

Front

What is the Central Limit Theorem?

Back

The Central Limit Theorem states that the distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the shape of the population distribution.