AP Stats [Ch.7-8]

What are the mean and standard deviation of a geometric random variable?

mean = 1/p standard deviation = ( √1-p ) / p

What are the mean and standard deviation of a binomial random variable?

mean = u = np standard deviation = o = √np(1-p) ONLY FOR BINOMIAL DISTRIS

What is the value of [ n! / (n-1)! ] ?

n

In the binomial distribution, what do parameters n and p represent?

n is the number of observations and p is the probability of success on any one observation.

As the number of trials n gets larger, the binomial distribution gets close to a _______

normal distribution

If X has a geometric distribution, what does (1 - p)n - 1p represent?

the probability that the first success occurs on the nth trial.

How do you calculate the mean of a discrete random variable X?

u = E(xi*pi)

What are the four conditions for the binomial setting?

1. Two categories "success" or "failure." 2. n observations are all independent. 3. Fixed number of n. 4. Probability of success same for each observation.

What are the four conditions for the geometric setting?

1. Two categories "success" or "failure." 2. n observations are all independent. 3. Variable of interest is the number of trials required to obtain 1st success 4. Probability of success same for each observation.

What is a continuous random variable?

A continuous random variable X takes all values in an interval of numbers.

What is a discrete random variable?

A discrete random variable X has a countable number of possible values.

What is the difference between a discrete random variable and a continuous random variable?

A discrete random variable has a countable number of possible values. A continuous random variable has an infinite number of possible values, all the values in an interval.

What is a random variable?

A random variable is a variable whose value is a numerical outcome of a random phenomenon.

What does the density curve of a uniform distribution look like?

A uniform distribution has the same height [ 1 / (b-a) ], over the entire interval (a, b.)

What are the two assumptions we much check to use the normal approximation to the binomial?

As a rule of thumb, we will use the normal approximation when n and p satisfy np > 10 and n(1-p) > 10

What is meant by B(n, p) ?

B(n, p) is the binomial distribution of the count X of successes in the binomial setting. The possible values of X are the whole numbers from 0 to n.

In a probability histogram what is the sum of the heights of each bar?

Because the heights are probabilities, they add to 1.

If X is a discrete random variable, do and have the same value? Explain.

No. P(X = 2) may be positive, so P(X ≥ 2) would equal P(X > 2) + P( X = 2).

How is a normal distribution related to probability distribution?

Normal distributions are one type of continuous probability distribution

What is the rule for calculating geometric probabilities? If X has a geometric distribution with probability of success p and failure (1 - p) on each observation, the possible values of X are 0, 1, 2, .. . If n is any one of these values, then the probability that the first success occurs on the nth trial is found by:

P(x = n) = (1-p)^(n-1)*p

What is the rule for calculating binomial probabilities? If X has the binomial distribution with n observations and probability of success p on each observation, the possible values of X are 0, 1, 2, .. ., n. If k is one of these values, what is P(X = k)?

P(x=k) = (n over k)*p^k*(1-p)^(n-k)

Suppose Ux = 2. According to the rules for means, what is U3+4x?

U3+4x = 3 + 4(2) = 11

What are the Rules for Means?

Ua+bx = a + bUx Ux+y = Ux + Uy Ux-y = Ux - Uy

What is the difference between a probability distribution function and a cumulative distribution function?

Given a discrete random variable X, the probability distribution function assigns a probability to each value of X. The cumulative distribution function of X calculates the sum of the probabilities for 0, 1, 2, ..., up to the value of X. That is, it calculates the probability of obtaining at most X successes in n trials.

Explain the difference between the binomial setting and the geometric setting

In the binomial setting we have a fixed number of observations and we count the number of successes in those n observations. In the geometric setting we are looking for how many trials we need to obtain the first success. The number of observations can be 1 to infinity

If a normal distribution is always a probability distribution, is a probability distribution always a normal distribution?

No, not every probability distribution is a normal distribution

Suppose Ox² = 4. According to the rules for variances, what is O3+2x²? What is O3+2x?

O3+2x² = 2² (4) = 16 O3+2x = √16 = 4

What are the Rules for Variances?

Oa+bx² = b²*ox² Oa+bx² = ox²*oy² Oa-bx² = ox²*oy²

Suppose Ox² = 2 and Oy² = 3 and X and Y are independent random variables. According to the rules for variances, what is Ox+y²? What is Ox+y?

O²x+y = 2 + 3 = 5 Ox+y = √5

Explain the Law of Large Numbers.

The Law of Large Numbers says that the average of the values of X observed in many trials must approach u

What is the expected value of a geometric random variable?

The expected value is the expected number of trials required to get the first success.

What is meant by the expected value of X ?

The expected value of a random variable X is an average of all possible values of X taking into account the fact that all values do not need to be equally likely. This expected value need not be a possible value for X.

In a probability histogram what does the height of each bar represent?

The height of each bar shows the probability of the outcome at its base

Explain the difference between the notations x-bar and Ux

The mean x-bar is the ordinary average of a set of observations. Ux is the mean, or expected value of a random variable X.

What is the correlation between two independent random variables?

Zero. If the variables are NOT independent then, O²x+/-y = O²x + O²y +/- 2pOxOy p = correlation between X and Y

If X is a discrete random variable, what information does the probability distribution of X give?

The probability distribution for X lists the values and their probabilities:

If X is a continuous random variable, how is the probability distribution of X described?

The probability distribution of X is described by a density curve.

What is the area under a probability density curve equal to?

The probability of any event is the area under the density curve and above the values of X that make up the event.

Given the variance of a random variable, explain how to calculate the standard deviation

The standard deviation equals the square root of the variance

Suppose Ux = 5 and Uy = 10. According to the rules for means, what is Ux+y?

Ux+y = 5 + 10 = 15

Does a linear combination of independent normal random variables have to be normally distributed?

Yes

If X is a continuous random variable, do and have the same value? Explain.

Yes, all continuous probability distributions assign probability 0 to every individual outcome. Only intervals of values have positive probability.

Variances of independent random variables _____; standard deviations ______.

add ; do not

Suppose that a count X has the binomial distribution with n trials and probability of success p. When n is large, the distribution of X is

approximately normal, N(np, √np(1-p))

In the formula ( n over k) = n! / k!(n-1)! , what does n represent? What does k represent? What does the value of (n over k) represent?

is the number of successes among n observations. k can be any whole numbers from 0 to n. ( n over k ) is the binomial coefficient n choose k, it is the number of ways of arranging k successes among n observations.