Continuous Random Variables and Probability Distributions

DR. SYED IMTIYAZ HASSAN

Assistant Professor, Department of CSE, Jamia Hamdard (Deemed To Be University), New Delhi, India.

https://syedimtiyazhassan.org, s.imtiyaz@jamiahamdard.ac.in

CONTINUOUS RANDOM VARIABLE

▪ A random variable with an interval (either finite or infinite) of real numbers for its range.

PROBABILITY DENSITY FUNCTION

A probability density function f(x) can be used to describe theprobability distribution of a continuous random variable X.

PROBABILITY DENSITY FUNCTION

▪ f(x) is used to calculate an area that represents the probability that X assumes a value in [a, b].

▪ A histogram is an approximation to a probability density function.

Example - 1

Let the continuous random variable X denote the current measured in a thin

copper wire in milliamperes. Assume that the range of X is [0, 20 mA], and

assume that the probability density function of X is f(x) = 0.05 for 0 ≤ x ≤ 20.

What is the probability that a current measurement is less than 10milliamperes?

CUMULATIVE DISTRIBUTION FUNCTIONS

MEAN and VARIANCE

CONTINUOUS UNIFORM DISTRIBUTION

NORMAL DISTRIBUTION

▪ The most widely used model for the distribution of a random variable.

▪ Also referred to as Gaussian distribution.

▪ It is a probability distribution that is symmetric about the mean.

▪ Data near the mean are more frequent in occurrence than data farfrom the mean.

▪ In graph form, normal distribution will appear as a bell curve.

NORMAL DISTRIBUTION

▪ The mean, mode and median are all equal.

▪ The curve is symmetric at the center (i.e. around the mean, μ).

▪ Exactly half of the values are to the left of center and exactly half the values are to the right.

NORMAL DISTRIBUTION

▪ 68% of the data falls within one standard deviation of the mean.

▪ 95% of the data falls within two standard deviations of the mean.

▪ 99.7% of the data falls within three standard deviations of the mean.

NORMAL DISTRIBUTION

▪ The probability density of the normal distribution is

STANDARD NORMAL DISTRIBUTION

STANDARD NORMAL DISTRIBUTION

Example - 2

Molly earned a score of 940 on a national achievement test. The mean test

score was 850 with a standard deviation of 100. What proportion of

students had a higher score than Molly?

(Assume that test scores are normally distributed.)

Z = (X - μ) / σ = (940 - 850) / 100 = 0.90

P(Z < 0.90) = 0.8159. (Z –Table)

Therefore, the P(Z > 0.90) = 1 - P(Z < 0.90) = 1 - 0.8159 = 0.1841.

Thus, we estimate that 18.41 percent of the students tested had a higher score

than Molly.

Example - 3

In a digital communication channel, assume that the number of bits received in

error can be modeled by a binomial random variable, and assume that the

probability that a bit is received in error is 1 x 10-5 . If 16 million bits are

transmitted, what is the probability that more than 150 errors occur?

Let the random variable X denote the number of errors.

NORMAL APPROXIMATION TO THE BINOMIAL DISTRIBUTIONS

Example – 3 (cont. . .)

In a digital communication channel, assume that the number of bits received inerror can be modeled by a binomial random variable, and assume that theprobability that a bit is received in error is 1 x 10-5 . If 16 million bits aretransmitted, what is the probability that more than 150 errors occur?

Let the random variable X denote the number of errors.

(Z –Table)

Example - 4

Assume that the number of asbestos particles in a squared meter of dust on asurface follows a Poisson distribution with a mean of 1000. If a squared meter ofdust is analyzed, what is the probability that less than 950 particles are found?

NORMAL APPROXIMATION TO THE POISSON DISTRIBUTIONS

Example – 4 (cont. . .)

Assume that the number of asbestos particles in a squared meter of dust on asurface follows a Poisson distribution with a mean of 1000. If a squared meter ofdust is analyzed, what is the probability that less than 950 particles are found?

EXPONENTIAL DISTRIBUTION

▪ Used to model the time elapsed between events.

▪ Probability distribution of the time between events in a Poissonpoint process.

▪ It is the continuous analogue of the geometric distribution

▪ Memoryless

▪ refers to the cases when the distribution of a "waiting time" until a certain event, does not depend on how much time has elapsed already.

▪ Only two kinds of distributions are memoryless: exponential distributions of non-negative real numbers and the geometric distributions of non-negative integers.

EXPONENTIAL DISTRIBUTION

ERLANG DISTRIBUTION

▪ A generalization of the exponential distribution.

▪ The random variable that equals the interval length until r counts occur in a Poisson process.

GAMMA DISTRIBUTION

▪ If the parameter r of an Erlang random variable is not an integer, but r > 0, the random variable has a gamma distribution.

▪ The Erlang distribution is a special case of the gamma distribution.

▪ Gamma function

▪ Γ(n)=(n−1)!

▪ General form

GAMMA DISTRIBUTION

OTHER DISTRIBUTIONS

▪ Chi-square distribution

▪ Student's t-distribution

▪ F distribution

▪ Weibull Distribution

▪ Beta distribution

▪ Log-normal distribution

EXPLORE

▪ Joint Probability Distribution

▪ Marginal Probability Distribution

▪ Conditional Probability Distribution

▪ Multinomial Probability Distribution

▪ Bivariate Normal Distribution

Thank YouThe probability of your possibilitydepends on level of your positivity.

Joseph Mercado