Let Xₙ be the random variable that equals the number of tails minus the number of heads when n fair coins are flipped. a) What is the expected value of Xₙ? b) What is the variance of Xₙ?

An employment evaluation exam has a variance of 250. Two particular exams with raw scores of 142 and 165 have $z$ scores of $-0.5$ and $0.955$, respectively. Determine the mean of the distribution.

Suppose ƒ is a probability density function for the random variable X with mean

$$\mu.$$

Show that its variance satisfies

$$\operatorname { Var } ( X ) = \int _ { - \infty } ^ { \infty } x ^ { 2 } f ( x ) d x - \mu ^ { 2 }.$$

A sample of $\operatorname{size} n=19$ has variance $s^2=1.96$. At $\alpha=.05$ in a right-tailed test, does this sample contradict the hypothesis that $\sigma^2=1.21$?

Scores on an examination are assumed to be normally distributed with mean 78 and variance 36. If it is known that a student’s score exceeds 72, what is the probability that his or her score exceeds 84?

For the probability density functions defined, find (a) the expected value, (b) the variance, (c) the standard deviation, (d) the median, and (e) the cumulative distribution function. $f(x)=\frac{1}{5} ;[4,9]$

Calculate the expected value, the variance, and the standard deviation of the given random variable X. (Round all answers to two decimal places.) X is the higher number when two dice are rolled.

Find the mean, median, and mode for each set of data. than find the range, variance, and standard deviation for each population. {1, 1, 1, 2, 2, 3}

Find the $95$ percent confidence interval for the population variance from these samples.

a. $n=15$ commuters, $s=10$ miles driven

b. $n=18$ students, $s=12$ study hours

Find the variance and standard deviation of the following sample

$$\begin{array}{llllllllll}0 & -5 & -3 & 6 & 4 & -4 & 1 & -5 & 0 & 3\end{array}$$

The English alphabet contains 21 consonants and five vowels. How many strings of six lowercase letters of the English alphabet contain a) exactly one vowel? b) exactly two vowels? c) at least one vowel? d) at least two vowels?

Decrypt these messages encrypted using the shift cipher f p) = (p + 10) mod 26. a) CEBBOXNOB XYG b) LO WI PBSOXN c) DSWO PYB PEX

Give a recursive definition of a) the set of odd positive integers. b) the set of positive integer powers of 3. c) the set of polynomials with integer coefficients.

In Exercises given below, identify the probability density function. Then find the mean, variance, and standard deviation without integrating.

$$f(x)=\frac{1}{8} e^{-x / 8},[0, \infty)$$