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César Jair Valles Sarmiento

César Jair V.

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Suppose a fair 6 -sided die is rolled six independent times. A match occurs if side $i$ is observed on the $i$ th trial, $i=1, \ldots, 6$. (a) What is the probability of at least one match on the six rolls? Hint: Let $C_{i}$ be the event of a match on the $i$ th trial and use Exercise $1.4 .13$ to determine the desired probability. (b) Extend part (a) to a fair $n$ -sided die with $n$ independent rolls. Then determine the limit of the probability as $n \rightarrow \infty$.

Suppose a fair 6 -sided die is rolled six independent times. A match occurs if side $i$ is observed on the $i$ th trial, $i=1, \ldots, 6$. (a) What is the probability of at least one match on the six rolls? Hint: Let $C_{i}$ be the event of a match on the $i$ th trial and use Exercise $1.4 .13$ to determine the desired probability. (b) Extend part (a) to a fair $n$ -sided die with $n$ independent rolls. Then determine the limit of the probability as $n \rightarrow \infty$.

Introduction to Mathematical Statistics

Probability and Distributions

Conditional Probability and Independence
A die is cast independently until the first 6 appears. If the casting stops on an odd number of times, Bob wins; otherwise, Joe wins. (a) Assuming the die is fair, what is the probability that Bob wins? (b) Let $p$ denote the probability of a 6 . Show that the game favors Bob, for all $p$, $0<p<1$

A die is cast independently until the first 6 appears. If the casting stops on an odd number of times, Bob wins; otherwise, Joe wins. (a) Assuming the die is fair, what is the probability that Bob wins? (b) Let $p$ denote the probability of a 6 . Show that the game favors Bob, for all $p$, $0<p<1$

Introduction to Mathematical Statistics

Probability and Distributions

Conditional Probability and Independence
A person answers each of two multiple choice questions at random. If there are four possible choices on each question, what is the conditional probability that both answers are correct given that at least one is correct?

A person answers each of two multiple choice questions at random. If there are four possible choices on each question, what is the conditional probability that both answers are correct given that at least one is correct?

Introduction to Mathematical Statistics

Probability and Distributions

Conditional Probability and Independence

Questions asked

ANSWERED

Luke Humphrey verified

Numerade educator

Let X be a continuous random variable with probability density function given by f(x)= -(1/2)x + 1, 0 ≤ x ≤ 2. (a) Graph the density function f (x). b) Find the total area under f (x) for 0 ≤ x ≤ 2. c) Find P(X≥1) using geometry and integration. d) Find P(X≤1/2). e) Find P(X = 1½). f) Find the expected value and variance of X. g) Find the cumulative distribution function of X.

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ANSWERED

Luke Humphrey verified

Numerade educator

Let X and Y be random variables with expected values μ =μ_x = μ_y and variances σ = (σ_x)^2 = (σ_y)^2. Let Z = (2X + Y) / 2. a) Find the expected value of Z. b) Find the variance of Z assuming X and Y are statistically independent. c) Find the variance of Z assuming that the correlation between X and Y is -0.5. d) Let the correlation between X and Y be -0.5. Find the correlation between aX and bY, where a and b are nonzero constants.

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INSTANT ANSWER

Suponga que X e Y tienen una distribución conjunta discreta para lo cual la f.d.a. se define como sigue: f(x,y) = (1 /30)(x + y) para x = 0,1,2, y = 0,1,2,3, 0 de lo contrario a. Determine las funciones marginales de X e Y. b. ¿Son X e Y independientes?

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ANSWERED

Keondre Parker verified

Numerade educator

egin{tabular}{|c|c|c|c|c|c|} hline & multicolumn{5}{|c|}{( mathrm{Y} )} \ hline ( mathrm{X} ) & 0 & 1 & 2 & 3 & 4 \ hline 0 & 0.08 & 0.07 & 0.06 & 0.01 & 0.01 \ hline 1 & 0.06 & 0.10 & 0.12 & 0.05 & 0.02 \ hline 2 & 0.05 & 0.06 & 0.09 & 0.04 & 0.03 \ hline 3 & 0.02 & 0.03 & 0.03 & 0.03 & 0.04 \ hline end{tabular}

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INSTANT ANSWER

Sea X una variable aleatoria continua con función de densidad de probabilidad dada por f(x)= −(1/2)x + 1, 0 ≤ x ≤ 2 a. Grafique la función de densidad f (x). b. Encuentre el área total debajo de f (x) para 0 ≤ x ≤ 2. c. Encuentre P(X≥1) usando geometría e integración. d. Encuentra P(X≤1/2) e. Encuentra P (X = 1½) f. Encuentre el valor esperado y la varianza de X. g. Encuentre la función de distribución acumulativa de X.

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INSTANT ANSWER

Sean X e Y variables aleatorias con valores esperados μ =μ_x = μ_y y varianzas σ = (σ_x)^2 = (σ_y)^2. Dejar Z = (2X + Y) / 2. a. Encuentre el valor esperado de Z. b. Encuentre la varianza de Z suponiendo que X e Y son estadísticamente independientes. c. Encuentre la varianza de Z suponiendo que la correlación entre X e Y es −0.5. d. Sea la correlación entre X e Y −0.5. Encuentre la correlación entre aX y bY, donde a y b son constantes distintas de cero.

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ANSWERED

James Kiss verified

Numerade educator

Consider the following three data sets A, B, and C. A = {9,10,11,7,13} B = {10,10,10,10,10} C = {1,1,10,19,19} a) Calculate the mean of each data set b) Calculate the standard deviation of each data set c) Which set has the largest standard deviation?

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ANSWERED

Audrey Fong verified

Numerade educator

Un vendedor de helados lleva 20 galones de helado en su camión todos los días. Sea X el número de galones que ella vende. La probabilidad es 0.1 de que X = 20. Si ella no vende los 20 galones, la distribución de X sigue una distribución continua con una f.d.p. de la forma: f(x)= {cxpara0 <x <20 0 de lo contrario donde c es una constante que hace que Pr (X <20) = 0.9. Encuentra la constante c de modo que Pr (X <20) = 0.9 como se describió anteriormente.

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ANSWERED

Robin Corrigan verified

Numerade educator

Suponga que la f.d.p. de una variable aleatoria X es como sigue: f (x) = c /(1 − x)1/2 para 0 <x <1, 0 en caso contrario. a. Encuentre el valor de la constante c y dibuje la f.d.p. b. Encuentre el valor de Pr (X ≤ 1/2).

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ANSWERED

Amman Zia verified

Numerade educator

La moda de distribución de una variable aleatoria X es un valor de x que maximiza la fdp o fmp. Si solo hay una x, se llama moda de la distribución. Encuentre la moda de cada una de las siguientes distribuciones: 1. f(x)= {12x2(1 − x)para0 <x <1 0 en cualquier otro lugar 2. f(x)= {(1/2)x2?−?,0 <x <∞,1 0 en otra parte

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