الرئيسية
Binomial Distribution
Binomial distributions model (some) discrete random variables.
Typically, a binomial random variable is the number of successes in a series of trials, for example, the number of 'heads' occurring when a coin is tossed 50 times.
A discrete random variable X is said to follow a Binomial distribution with parameters n and p, written X ~ Bi(n,p) or X ~ B(n,p), if it has probability distribution
where
x = 0, 1, 2, ......., n
n = 1, 2, 3, .......
p = success probability; 0 < p < 1
The trials must meet the following requirements:
a. the total number of trials is fixed in advance;
b. there are just two outcomes of each trial; success and failure;
c. the outcomes of all the trials are statistically independent;
d. all the trials have the same probability of success.
The Binomial distribution has expected value E(X) = np and variance V(X) = np(1-p).
Examples
Probability Mass Function The binomial distribution is used when there are exactly two mutually exclusive outcomes of a trial. These outcomes are appropriately labeled "success" and "failure". The binomial distribution is used to obtain the probability of observing x successes in N trials, with the probability of success on a single trial denoted by p. The binomial distribution assumes that p is fixed for all trials.
The formula for the binomial probability mass function is
where
Simple Example
The four possible outcomes that could occur if you flipped a coin twice are listed below in Table 1. Note that the four outcomes are equally likely: each has probability 1/4. To see this, note that the tosses of the coin are independent (neither affects the other). Hence, the probability of a head on Flip 1 and a head on Flip 2 is the product of P(H) and P(H), which is 1/2 x 1/2 = 1/4. The same calculation applies to the probability of a head on Flip one and a tail on Flip 2. Each is 1/2 x 1/2 = 1/4.
Table 1. Four Possible Outcomes.
Outcome First Flip Second Flip
1 Heads Heads
2 Heads Tails
3 Tails Heads
4 Tails Tails
The four possible outcomes can be classified in terms of the number of heads that come up. The number could be two (Outcome 1), one (Outcomes 2 and 3) or 0 (Outcome 4). The probabilities of these possibilities are shown in Table 2 and in Figure 1. Since two of the outcomes represent the case in which just one head appears in the two tosses, the probability of this event is equal to 1/4 + 1/4 = 1/2. Table 1 summarizes the situation.
Table 2. Probabilities of Getting 0, 1, or 2 heads.
Number of Heads Probability
0 1/4
1 1/2
2 1/4
Figure 1. Probabilities of 0, 1, and 2 heads.
Figure 1 is a discrete probability distribution: It shows the probability for each of the values on the X-axis. Defining a head as a "success," Figure 1 shows the probability of 0, 1, and 2 successes for two trials (flips) for an event that has a probability of 0.5 of being a success on each trial. This makes Figure 1 an example of a binomial distribution
Binomial distributions model (some) discrete random variables.
Typically, a binomial random variable is the number of successes in a series of trials, for example, the number of 'heads' occurring when a coin is tossed 50 times.
A discrete random variable X is said to follow a Binomial distribution with parameters n and p, written X ~ Bi(n,p) or X ~ B(n,p), if it has probability distribution
where
x = 0, 1, 2, ......., n
n = 1, 2, 3, .......
p = success probability; 0 < p < 1
The trials must meet the following requirements:
a. the total number of trials is fixed in advance;
b. there are just two outcomes of each trial; success and failure;
c. the outcomes of all the trials are statistically independent;
d. all the trials have the same probability of success.
The Binomial distribution has expected value E(X) = np and variance V(X) = np(1-p).
Examples
Probability Mass Function The binomial distribution is used when there are exactly two mutually exclusive outcomes of a trial. These outcomes are appropriately labeled "success" and "failure". The binomial distribution is used to obtain the probability of observing x successes in N trials, with the probability of success on a single trial denoted by p. The binomial distribution assumes that p is fixed for all trials.
The formula for the binomial probability mass function is
where
Simple Example
The four possible outcomes that could occur if you flipped a coin twice are listed below in Table 1. Note that the four outcomes are equally likely: each has probability 1/4. To see this, note that the tosses of the coin are independent (neither affects the other). Hence, the probability of a head on Flip 1 and a head on Flip 2 is the product of P(H) and P(H), which is 1/2 x 1/2 = 1/4. The same calculation applies to the probability of a head on Flip one and a tail on Flip 2. Each is 1/2 x 1/2 = 1/4.
Table 1. Four Possible Outcomes.
Outcome First Flip Second Flip
1 Heads Heads
2 Heads Tails
3 Tails Heads
4 Tails Tails
The four possible outcomes can be classified in terms of the number of heads that come up. The number could be two (Outcome 1), one (Outcomes 2 and 3) or 0 (Outcome 4). The probabilities of these possibilities are shown in Table 2 and in Figure 1. Since two of the outcomes represent the case in which just one head appears in the two tosses, the probability of this event is equal to 1/4 + 1/4 = 1/2. Table 1 summarizes the situation.
Table 2. Probabilities of Getting 0, 1, or 2 heads.
Number of Heads Probability
0 1/4
1 1/2
2 1/4
Figure 1. Probabilities of 0, 1, and 2 heads.
Figure 1 is a discrete probability distribution: It shows the probability for each of the values on the X-axis. Defining a head as a "success," Figure 1 shows the probability of 0, 1, and 2 successes for two trials (flips) for an event that has a probability of 0.5 of being a success on each trial. This makes Figure 1 an example of a binomial distribution
» 77 كتاب للطاقة الداخلية + 9 كتب للباراسيكولوجى
» حصريا فيلم ((ابراهيم الابيض)) بطولة احمد السقا و هند صبرى نسخة أصلية Rmvb حجم 380 MB
» افلام (فيديوهات) اللعبه الشهيرة need for speed most wanted
» مكتبة الاسكندرية صرح التاريخ(المكتبة الحديثة)
» مكتبة الاسكندرية صرح التاريخ(اللمكتبة القديمة)
» لعبة الصراحة فى الحب
» صور تحت المطر لازم تتفرج عليها
» فلسطين في موسوعة جينيس
» اصدقاء