Answered: In this problem, we will approximate a…

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

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Please help with problems a, b, and c:

In this problem, we will approximate a binomial dis- tribution by a normal distribution. From looking at the pmfs of the binomial distribution, for large values of n the histogram of the binomial pmf looks “bell-shaped”. It turns out that the normal distribution is a good approximation for the binomial approximation for large n. It also works for small n as long as p is not too far from 0.5. For convenience, we will let q = 1 − p.

(a) Let \( X \sim \text{Binomial}(25, 0.4) \). Compute \( P(X = 10) \).

(b) We will approximate \( X \) with a normal random variable. Let \( Y \) be a normal random variable with the same mean and variance as \( X \). Compute \( P(9.5 \leq Y \leq 10.5) \). We will use this as an approximation for \( P(X = 10) \).

(c) Briefly explain why we used the event \( (9.5 \leq Y \leq 10.5) \) to approximate the event \( (X = 10) \).

Features Features Normal distribution is characterized by two parameters, mean (µ) and standard deviation (σ). When graphed, the mean represents the center of the bell curve and the graph is perfectly symmetric about the center. The mean, median, and mode are all equal for a normal distribution. The standard deviation measures the data's spread from the center. The higher the standard deviation, the more the data is spread out and the flatter the bell curve looks. Variance is another commonly used measure of the spread of the distribution and is equal to the square of the standard deviation.

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