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Chapter 15: Q5P (page 765)

Computer plot on the same axes the normal density functions with μ=0σ=1and, 2, and 5. Label each curve with itsσ.

Short Answer

Expert verified

The first, second and third curves are symmetric across the vertical axis but have different limits.

The graph is given below

Step by step solution

01

Given information

  • The normal probability density function with μ=0,σ=1, μ=0,σ=2and μ=0,σ=3
02

Definition of probability density function

A probability density function (PDF) is a statistical expression that describes a discrete random variable's probability distribution.

03

Draw the graph of given functions

The graph is given below

The first, second and third curves are symmetric across the vertical axis but have different limits.

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Most popular questions from this chapter

(a) Find the probability that in two tosses of a coin, one is heads and one tails. That in six tosses of a die, all six of the faces show up. That in12tosses of a12 -sided die, all12 faces show up. That in n tosses of an n-sided die, all n faces show up.

(b) The last problem in part (a) is equivalent to finding the probability that, when n balls are distributed at random into n boxes, each box contains exactly one ball. Show that for large n, this is approximatelye−n2πn.

Two dice are thrown. Use the sample space (2.4) to answer the following questions.

(a) What is the probability of being able to form a two-digit number greater than

33 with the two numbers on the dice? (Note that the sample point 1, 4 yields

the two-digit number 41 which is greater than 33, etc.)

(b) Repeat part (a) for the probability of being able to form a two-digit number

greater than or equal to 42.

(c) Can you find a two-digit number (or numbers) such that the probability of

being able to form a larger number is the same as the probability of being able

to form a smaller number? [See note part (a)]

Set up an appropriate sample space for each of Problems 1.1 to 1.10 and use itto solve the problem. Use either a uniform or non-uniform sample space or try both.

If you select a three-digit number at random, what is the probability that the units digit is 7? What is the probability that the hundreds digit is 7?

Given a non uniform sample space and the probabilities associated with the points, we defined the probability of an event A as the sum of the probabilities associated with the sample points favorable to A. [You used this definition in Problem 15with the sample space (2.5).] Show that this definition is consistent with the definition by equally likely cases if there is also a uniform sample space for the problem (as there was in Problem 15). Hint: Let the uniform sample space have n<Npoints each with the probability N-1. Let the nonuniform sample space have n points, the first point corresponding to N1 points of the uniform space, the second to N2 points, etc. What is N1 + N2 + .... Nn ?What are p1, p2, ...the probabilities associated with the first, second, etc., points of the nonuniform space? What is p1 + p2 +....+ pn? Now consider an event for which several points, say i, j, k, of the nonuniform sample space are favorable. Then using the nonuniform sample space, we have, by definition of the probability p of the event, p = pi + pj + pk . Write this in terms of the N’s and show that the result is the same as that obtained by equally likely cases using the uniform space. Refer to Problem 15as a specific example if you need to.

Two cards are drawn at random from a shuffled deck.

  1. What is the probability that at least one is a heart?

(b) If you know that at least one is a heart, what is the probability that both are

hearts?
See all solutions

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