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In some situationsX is continuous but Y is discrete. For example, in a logistic regression, one may wish to predict the probability of a binary outcome Y conditional on the value of a continuously-distributed X. In this case, (X, Y) has neither a probability density function nor a probability mass function in the sense of the terms given above. On the other hand, a "mixed joint density" can be defined in either of two ways:

Formally, fX,Y(x, y) is the probability density function of (X, Y) with respect to the product measure on the respective supports of X and Y. Either of these two decompositions can then be used to recover the joint cumulative distribution function:

The definition generalizes to a mixture of arbitrary numbers of discrete and continuous random variables.

Q: Discuss distribution function of mixed random variable?

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The answer depends on the probability distribution function for the random variable.

Almost all statistical distribution have a mean. It is the expected value of the random variable which is distributed according to that function.

I have included two links. A normal random variable is a random variable whose associated probability distribution is the normal probability distribution. By definition, a random variable has to have an associated distribution. The normal distribution (probability density function) is defined by a mathematical formula with a mean and standard deviation as parameters. The normal distribution is ofter called a bell-shaped curve, because of its symmetrical shape. It is not the only symmetrical distribution. The two links should provide more information beyond this simple definition.

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The probability of a random variable being at or below a certain value is defined as the cumulative distribution function (CDF) of the variable. The CDF gives the probability that the variable takes on a value less than or equal to a given value.

Related questions

You integrate the probability distribution function to get the cumulative distribution function (cdf). Then find the value of the random variable for which cdf = 0.5.

The marginal probability distribution function.

The answer depends on the probability distribution function for the random variable.

A probability density function can be plotted for a single random variable.

A random variable is a variable that can take different values according to a process, at least part of which is random.For a discrete random variable (RV), a probability distribution is a function that assigns, to each value of the RV, the probability that the RV takes that value.The probability of a continuous RV taking any specificvalue is always 0 and the distribution is a density function such that the probability of the RV taking a value between x and y is the area under the distribution function between x and y.

The value of the distribution for any value of the random variable must be in the range [0, 1]. The sum (or integral) of the probability distribution function over all possible values of the random variable must be 1.

Almost all statistical distribution have a mean. It is the expected value of the random variable which is distributed according to that function.

I have included two links. A normal random variable is a random variable whose associated probability distribution is the normal probability distribution. By definition, a random variable has to have an associated distribution. The normal distribution (probability density function) is defined by a mathematical formula with a mean and standard deviation as parameters. The normal distribution is ofter called a bell-shaped curve, because of its symmetrical shape. It is not the only symmetrical distribution. The two links should provide more information beyond this simple definition.

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It depends on what the random variable is, what its domain is, what its probability distribution function is. The probability that a randomly selected random variable has a value between 40 and 60 is probably quite close to zero.

It is a function that gives the probabilities associated with the discrete number of values that a random variable can take.

Cauchy distribution is the distribution of a random variable along a specific function. In AI, this distribution is used to generate adaptive models which produce fast learning across dimensions.