Let Z have a standard normal distribution and define a new rv ?Y ?by Y = ? Z + ? . Show that ?Y ?has a normal distribution with parameters ? and ? . [?Hint?: Y ? y iff Z ? ? Use this to find the cdf of ?Y ?and then differentiate it with respect to ?y ?.]
Statistics 201 – Professor Baek Section Titles Vocab Subtitles Chapter 6: Probability Distributions Section 6.1 1. Random Variable: a numerical measurement of the outcome of a random phenomenon; randomness often results from the use of random sampling or a randomized experiment to gather the data 2. Probability Distribution: specifies its possible values and their probabilities (for random variables) 3. Probability distribution of a discrete random variable assigns a probability to each possible value a. For each x, the probability P(x) falls between 0 and 1 b. The sum of the probabilities for all the possible x values equals 1 4. The mean of a probability distribution for a discrete random variable is: a. = x P(x) b. called a weighted average – used when each x value is not equally likely; if a particular x value is more likely to occur, it has a larger influence on the mean, which is the balance point of the distribution c. also called the expected value of X – observes what we expect for the average in a long run of observations 5. the standard deviation of a probability distribution, denoted by , measures the variability from the mean 6. A continuous random variable has possible values that form an interval a. Its probability distribution is specified by a curve that determines the probability that the random variable falls in any particular interval o
