A continuous random variable X takes all values in an interval of numbers. The probability distribution of X is described by a density curve. The probability of any event is the area under the density curve and above the values of X that make up the event.
The probability distribution for a continuous random variable assigns probabilities to intervals of outcomes rather than to individual outcomes. All continuous probability models assign probability 0 to every individual outcome. Only intervals of values have positive probability. This is because when we measure the area under the curve, a single value has no width, so the area is 0. Since that is the case, P(0.3≤Y≤0.7) = P(0.3<Y<0.7)
If we want to know which random variable (if either) has the higher expected gain in the long run or which has the larger variability, we need to investigate random variables further.
When adding or subtracting a constant:
- Adds to measures of center and location
- Mean, median, quartiles, percentiles
- Does not change shape or measure of spread
- Range, IQR, Standard Deviation When multiplying or dividing a constant:
- Multiplies measures of center and location
- Mean, median, quartiles, and percentiles
- Multiplies measures of spread
- Range, IQR, standard deviation
- Multiplies the variance by the value squared
- Does not change the shape of the distribution
If we multiply a random variable by a negative constant b, the measures of spread would be multiplied by |b| (the absolute value of b). If we multiply a random variable by constant b, the variance would be multiplied by b^2.
If Y = a+bX is a linear transformation of the random variable X, then the probability distribution of Y has the same shape as the probability distribution of X if b > 0. Uy = a + bUx and σy = |b|Ox (because b could be negative).
Mean of the sum of random variables
For any two random variables X and Y, if T = X + Y, then the expected value of T is E(T) = μt = μx̅ + Uy
In general, the mean of the sum of several random variables is the sum of their means.
Because more variables mean more variability, we need to look at how we can calculate measures of variability. The range of T = range of X + range of Y
Then, we need to remember that we can’t calculate the probability for any value of T unless X and Y are independent random variables.
Independent random variables
If knowing whether any event involving X alone has occurred tells us nothing about the occurrence of any event involving Y alone and vice versa, then X and Y are independent random variables.
Probability models often assume independence when the random variables describe outcomes that appear unrelated to each other. You should always ask whether the assumption of independence seems reasonable.
Important note: When we add two independent random variables, their variances add. Standard deviations do not add.
For any two independent random variables X and Y, if T = X + Y, then the variance of T is σt^2 = σx^2 + σy^2
In general, the variance of the sum of several independent random variables is the sum of their variances.
Mean of the difference of random variables
For any two random variables X and Y, if D = X - Y, then the mean of D is µd = E(D) = μx̅ - µy
Variance of the difference of random variables
For any two independent random variables X and Y, if D = X - Y, then the variance of D is σd^2 = σx^2 + σy^2 Remember, this stays addition, the same as when you find the variance for the sum of random variables.