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Chapter 3: Q7E (page 100)

Suppose that a random variableXhas the binomial distribution

with parametersn=8 andp=0.7. Find Pr(X鈮�5)by using the table given at the end of this book. Hint: Use the fact that Pr(X鈮�5)=Pr(Y鈮�3), whereYhas thebinomial distribution with parametersn=8 andp=0.3.

Short Answer

Expert verified

The probability that X is greater than or equal to 5 is 0.8059.

Step by step solution

01

Given information

The random variable X follows the binomial distribution with parameters \(n = 8\) and \(p = 0.7\)

02

Obtain the required probability in terms of variable Y   

Consider Y as\({\bf{Y = 1 - X}}\) .

Therefore, Y follows binomial distribution with,

\(\begin{aligned}{c}n& = 8\\q &= 1 - p\\ &= 1 - 0.7\\& = 0.3\end{aligned}\)

Thus, \(Y \sim Bin\left( {8,0.3} \right)\) .

Therefore, the required probability is transformed as,

\(P\left( {X \ge 5} \right) = P\left( {Y \le 3} \right)\)

03

Compute the probability using the table  

The required probability is computed as,

\[\begin{aligned}{c}P\left( {X \ge 5} \right) &= P\left( {Y \le 3} \right)\\ &= P\left( {Y = 0} \right) + P\left( {Y = 1} \right) + ... + P\left( {Y = 3} \right)\\ &= 0.0576 + 0.1977 + 0.2965 + 0.2541\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {{\rm{Using}}\;{\rm{Table}}\;{\rm{of}}\;{\rm{Binomial}}\;{\rm{Probabilities}}} \right)\\ \approx 0.8059\end{aligned}\]

Therefore, the required probability is 0.8059.

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Most popular questions from this chapter

Suppose that the joint distribution of X and Y is uniform over a set A in the xy-plane. For which of the following sets A are X and Y independent?

a. A circle with a radius of 1 and with its center at the origin

b. A circle with a radius of 1 and with its center at the point (3,5)

c. A square with vertices at the four points (1,1), (1,鈭�1), (鈭�1,鈭�1), and (鈭�1,1)

d. A rectangle with vertices at the four points (0,0), (0,3), (1,3), and (1,0)

e. A square with vertices at the four points (0,0), (1,1),(0,2), and (鈭�1,1)

Question:Suppose thatXandYhave a continuous joint distribution for which the joint p.d.f. is

\({\bf{f}}\left( {{\bf{x,y}}} \right){\bf{ = }}\left\{ \begin{array}{l}{\bf{k}}\;{\bf{for}}\;{\bf{a}} \le {\bf{x}} \le {\bf{b}}\;{\bf{and}}\;{\bf{c}} \le {\bf{y}} \le {\bf{d}}\\{\bf{0}}\;{\bf{otherwise}}\end{array} \right.\)

wherea <b,c < d, andk >0.

Find the marginal distributions ofXandY.

Suppose that \({{\bf{X}}_{\bf{1}}}{\bf{ \ldots }}{{\bf{X}}_{\bf{n}}}\) form a random sample of sizen from the uniform distribution on the interval [0, 1] andthat \({{\bf{Y}}_{\bf{n}}}{\bf{ = max}}\left( {{{\bf{X}}_{\bf{1}}}{\bf{ \ldots }}{{\bf{X}}_{\bf{n}}}} \right)\). Find the smallest value of \({\bf{n}}\)such that\({\bf{Pr}}\left( {{{\bf{Y}}_{\bf{n}}} \ge {\bf{0}}{\bf{.99}}} \right) \ge {\bf{0}}{\bf{.95}}\).

Question:For the joint pdf in example 3.4.7,determine whether or not X and Y are independent.

Each time that a shopper purchases a tube of toothpaste, she chooses either brand A or brand B. Suppose that the probability is 1/3 that she will choose the same brand chosen on her previous purchase, and the probability is 2/3 that she will switch brands.

a. If her first purchase is brand A, what is the probability that her fifth purchase will be brand B?

b. If her first purchase is brand B, what is the probability that her fifth purchase will be brand B?
See all solutions

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